Newton famously declared that it is not the business of science to make hypotheses. However, it's well to remember that this position was formulated in the midst of a bitter dispute with Robert Hooke, who had criticized Newton's writings on optics when they were first communicated to the Royal Society in the early 1670's. The essence of Newton's thesis was that white light is composed of a mixture of light of different elementary colors, ranging across the visible spectrum, which he had demonstrated by decomposing white light into its separate colors and then reassembling those components to produce white light again. However, in his description of the phenomena of color Newton originally included some remarks about his corpuscular conception of light (perhaps akin to the cogs and flywheels in terms of which James Maxwell was later to conceive of the phenomena of electromagnetism). Hooke interpreted the whole of Newton's optical work as an attempt to legitimize this corpuscular hypothesis, and countered with various objections.
Newton quickly realized his mistake in attaching his theory of colors to any particular hypothesis on the fundamental nature of light, and immediately back-tracked, arguing that his intent had been only to describe the observable phenomena, without regard to any hypotheses as to the cause of the phenomena. Hooke (and others) continued to criticize Newton's theory of colors by arguing against the corpuscular hypothesis, causing Newton to respond more and more angrily that he was making no hypothesis, he was describing the way things are, and not claiming to explain why they are. This was a bitter lesson for Newton and, in addition to initiating a life-long feud with Hooke, went a long way toward shaping Newton's rhetoric about what science should be.
I use the term "rhetoric" because it is to some extent a matter of semantics as to whether a descriptive theory entails a causative hypothesis. For example, when accused of invoking an occult phenomena in gravity, Newton replied that the phenomena of gravity are not occult, although the causes may be. (See below.) Clearly the dispute with Hooke had caused Newton to paint himself into the "hypotheses non fingo" corner, and this somewhat accidentally became part of his legacy to science, which has ever after been much more descriptive and less explanatory than, say, Descartes would have wished. This is particularly ironic in view of the fact that Newton personally entertained a great many bold hypotheses, including a number of semi-mystical hermetic explanations for all manner of things, not to mention his painstaking interpretations of biblical prophecies. Most of these he kept to himself, but when he finally got around to publishing his optical papers (after Hooke had died) he couldn't resist including a list of 31 "Queries" concerning the big cosmic issues that he had been too reticent to address publicly before. The true nature of these "queries" can immediately be gathered from the fact that every one of them is phrased in the form of a negative question, as in "Are not the Rays of Light very small bodies emitted from shining substances?" Each one is plainly a hypothesis phrased as a question.
The first edition of The Opticks (1704) contained only 16 queries, but when the Latin edition was published in 1706 Newton was emboldened to add seven more, which ultimately became Queries 25 through 31 when, in the second English edition, he added Queries 17 through 24. Of all these, one of the most intriguing is Query 28, which begins with the rhetorical question "Are not all Hypotheses erroneous in which Light is supposed to consist of Pression or Motion propagated through a fluid medium?" In this query Newton rejects the Cartesian idea of a material substance filling in and comprising the space between particles. Newton preferred an atomistic view, believing that all substances were comprised of hard impenetrable particles moving and interacting via innate forces in an empty space (as described further in Query 31). After listing several facts that make an aetheral medium inconsistent with observations, the discussion of Query 28 continues
And for rejecting such a medium, we have the authority of those the oldest and most celebrated philosophers of ancient Greece and Phoenicia, who made a vacuum and atoms and the gravity of atoms the first principles of their philosophy, tacitly attributing gravity to some other cause than dense matter. Later philosophers banish the consideration of such a cause... feigning [instead] hypotheses for explaining all things mechanically [But] the main business of natural philosophy is to argue from phenomena without feigning hypotheses, and to deduce causes from effects, till we come to the very first cause, which certainly is not mechanical.
And not only to unfold the mechanism of the world, but chiefly to resolve such questions as What is there in places empty of matter? and Whence is it that the sun and planets gravitate toward one another without dense matter between them? Whence is it that Nature doth nothing in vain? and Whence arises all that order and beauty which we see in the world? To what end are comets? and Whence is it that planets move all one and the same way in orbs concentrick, while comets move all manner of ways in orbs very excentrick? and What hinders the fixed stars from falling upon one another?
It's interesting to compare these comments of Newton with those of Socrates as recorded in Plato's Phaedo
If then one wished to know the cause of each thing, why it comes to be or perishes or exists, one had to find what was the best way for it to be, or to be acted upon, or to act. I was ready to find out ... about the sun and the moon and the other heavenly bodies, about their relative speed, their turnings, and whatever else happened to them, how it is best that each should act or be acted upon. I never thought [we would need to] bring in any other cause for them than that it was best for them to be as they are.
This wonderful hope was dashed as I went on reading, and saw that [men] mention as causes air and ether and water and many other strange things... It is what the majority appear to do, like people groping in the dark; they call it a cause, thus giving it a name which does not belong to it. That is why one man surrounds the earth with a vortex to make the heavens keep it in place, another makes the air support it like a wide lid. As for their capacity of being in the best place they could possibly be put, this they do not look for, nor do they believe it to have any divine force, but they believe that they will some time discover a stronger and more immortal Atlas to hold everything together...
Both men are suggesting that a hierarchy of mechanical causes cannot ultimately prove satisfactory, and that the first cause of things cannot be mechanistic in nature. Both suggest that the macroscopic mechanisms of the world are just manifestations of an underlying and irreducible principle of "order and beauty", indeed of a "divine force". But Newton wasn't content to leave it at this. After lengthy deliberations, and discussions with David Gregory, he decided to add the comment
Is not Infinite Space the Sensorium of a Being incorporeal, living and intelligent, who sees the things themselves intimately, and thoroughly perceives them, and comprehends them wholly by their immediate presence to himself?
Samuel Johnson once recommended a proof-reading technique to a young writer, telling him that you should read over your work carefully, and whenever you come across a phrase or passage that seems particularly fine, strike it out. Newton's literal identification of Infinite Space with the Sensorium of God may have been a candidate for that treatment, but it went to press anyway. However, as soon as the edition was released, Newton suddenly got cold feet, and realized that he'd exposed himself to ridicule. He desperately tried to recall the book and, failing that, he personally rounded up all the copies he could find, cut out the offending passage with scissors, and pasted in a new version. Hence the official versions contain the gentler statement (reverting once again to the negative question!):
And these things being rightly dispatch'd, does it not appear from phaenomena that there is a Being incorporeal, living, intelligent, omnipresent, who in infinite space, as it were in his Sensory, sees the things themselves intimately, and thoroughly perceives them, and comprehends them wholly by their immediate presence to himself: Of which things the images only carried through the organs of sense into our little sensoriums are there seen and beheld by that which in us perceives and thinks. And though every true step made in this philosophy brings us not immediately to the knowledge of the first cause, yet it brings us nearer to it...
Incidentally, despite Newton's efforts to prevent it, one of the un-repaired copies had already made its way out of the county, and was on its way to Leibniz, who predictably cited the original "Sensorium of God" comment as evidence that Newton "has little success with metaphysics".
Newton's 29th Query (not a hypothesis, mind you) was: "Are not the rays of light very small bodies emitted from shining substances?" Considering that his mooting of this idea over thirty years earlier had precipitated a controversy that nearly led him to a nervous breakdown, one has to say that Newton was nothing if not tenacious. This query also demonstrates how little his basic ideas about the nature of light had changed over the course of his life. After listing numerous reasons for suspecting that the answer to this question was Yes, Newton proceeded in Query 30 to ask the pregnant question "Are not gross bodies and light convertible into one another?" Following Newton's rhetorical device, should not this be interpreted as a suggestion of equivalence between mass and energy?
The final pages of The Opticks are devoted to Query 31, which begins
Have not the small particles of bodies certain powers, virtues, or forces, by which they act at a distance, not only upon the rays of light for reflecting, refracting, and inflecting them, but also upon one another for producing a great part of the Phenomena of nature?
Newton goes on to speculate that the force of electricity operates on very small scales to hold the parts of chemicals together and govern their interactions, anticipating the modern theory of chemistry. Most of this Query is devoted to an extensive (20 pages!) enumeration of chemical phenomena that Newton wished to cite in support of this view. He then returns to the behavior of macroscopic objects, asserting that
Nature will be very conformable to herself, and very simple, performing all the great motions of the heavenly bodies by the attraction of gravity which intercedes those bodies, and almost all the small ones of their particles by some other attractive and repelling powers which intercede the particles.
This is a very clear expression of Newton's belief that forces act between separate particles, i.e., at a distance. He continues
The Vis inertiae is a passive Principle by which Bodies persist in their Motion or Rest, receive Motion in proportion to the Force impressing it, and resist as much as they are resisted. By this Principle alone there never could have been any Motion in the World. Some other Principle was necessary for putting Bodies into Motion; and now they are in Motion, some other Principle is necessary for conserving the motion.
In other words, Newton is arguing that the principle of inertia, by itself, cannot account for the motion we observe in the world, because inertia only tends to preserve existing states of motion, and only uniform motion in a straight line. Thus we must account for the initial states of motion (the initial conditions), the persistence of non-inertial motions, and for the on-going variations in the amount of motion that are observed. For this purpose Newton distinguishes between "passive" attributes of bodies, such as inertia, and "active" attributes of bodies, such as gravity, and he points out that, were it not for gravity, the planets would not remain in their orbits, etc, so it is necessary for bodies to possess active as well as passive attributes, because otherwise everything would soon be diffuse and cold. Thus he is not saying that the planets would simply come to a halt in the absence of active attributes, but rather that the constituents of any physical universe resembling ours (containing persistent non-inertial motion) must necessarily possess active as well as passive properties.
Next, Newton argues that the "amount of motion" in the world is not constant, in two different respects. The first is rather interesting, because it makes very clear the fact that he regarded ontological motion as absolute. He considers two identical globes in empty space attached by a slender rod and revolving with angular speed w about their combined center of mass, and he says the center of mass is moving with some velocity v (in the plane of revolution). If the radius from the center of mass to each globe is r, then the globes have a speed of w r relative to the center. When the connecting rod is periodically oriented perpendicular to the velocity of the center, one of the globes has a speed equal to v + w r and the other a speed equal to v - w r, so the total "amount of motion" (i.e., the sum of the magnitudes of the momentums) is simply 2mv. However, when the rod is periodically aligned parallel to the velocity of the center, the globes each have a total speed of 
, so the total "amount of motion" is
Thus, Newton argues, the total quantity of motion of the two globes fluctuates periodically between this value and 2mv. Obviously he is expressing the belief that the "amount of motion" has absolute significance. (He doesn't remark on the fact that the kinetic energy in this situation is conserved).
The other way in which, Newton argues, the amount of motion is not conserved is in inelastic collisions, such as when two masses of clay collide and the bodies stick together. Of course, even in this case the momentum vector is conserved, but again the sum of the magnitudes of the individual momentums is reduced. Also, in this case, the kinetic energy is dissipated as heat. Interestingly, Newton observes that, aside from the periodic fluctuations such as with the revolving globes, the net secular change in total "amount of motion" is always negative.
By reason of the tenacity of fluids, the attrition of their parts... motion is much more apt to be lost than got, and is always upon the Decay.
This can easily be seen as an early statement of statistical thermodynamics and the law of entropy. In any case, from this tendency for motion to decay, Newton concludes that eventually the Universe must "run down", and "all things would grow cold and freeze, and become inactive masses".
Newton also mentions one further sense in which (he believed) passive attributes alone were insufficient to account for the persistence of well-ordered motion that we observe.
...blind fate could never make all the planets move one and the same way in orbs concentrick, some inconsiderable irregularities excepted, which may have risen from the action of comets and planets upon one another, and which will be apt to increase, till this system wants a reformation.
In addition to whatever sense of design and/or purpose we may discern in the initial conditions of the solar system, Newton also seems to be hinting at the idea that, in the long run, any initial irregularities, however "inconsiderable" they may be, will increase until the system wants reformation. In recent years we've gained a better appreciation of the fact that Newton's laws, though strictly deterministic, are nevertheless potentially chaotic, so that the overall long-term course of events can quickly come to depend on arbitrarily slight variations in initial conditions, rendering the results unpredictable on the basis of any fixed level of precision.
So, for all these reasons, Newton argues that passive principles such as inertia cannot suffice to account for what we observe. We also require active principles, among which he includes gravity, electricity, and magnetism. Beyond this, Newton suggests that the ultimate "active principle" underlying all the order and beauty we find in the world, is God, who not only set things in motion, but from time to time must actively intervene to restore their motion. This was an important point for Newton, because he was genuinely concerned about the moral implications of a scientific theory that explained everything as the inevitable consequence of mechanical principles. This is why he labored so hard to reconcile his clockwork universe with an on-going active role for God. He seems to have found this role in the task of resisting an inevitable inclination of our mechanisms to descend into dissipation and veer into chaos.
In this final Query Newton also took the opportunity to explicitly defend his abstract principles such as inertia and gravity, which some critics charged were occult.
These principles I consider not as occult qualities...but as general laws of nature, by which the things themselves are formed, their truth appearing to us by phenomena, though their causes be not yet discovered. For these are manifest qualities, and their causes only are occult. The Aristotelians gave the name of occult qualities not to manifest qualities, but to such qualities only as they supposed to lie hid in Bodies, and to be the unknown causes of manifest effects, such as would be the causes of gravity... if we should suppose that these forces or actions arose from qualities unknown to us, and uncapable of being made known and manifest. Such occult qualities put a stop to the improvement of natural philosophy, and therefore of late years have been rejected. To tell us that every species of things is endowed with an occult specific quality by which it acts and produces manifest effects is to tell us nothing...
The last set of Queries to be added, now numbered 17 through 24, appeared in the second English edition in 1717, when Newton was 75. These are remarkable in that they argue for an aether permeating all of space - despite the fact that Queries 25 through 31 argue at length against the necessity for an aether, and those were hardly altered at all when Newton added the new Queries which advocate an aether. (It may be worth noting, however, that the reference to "empty space" in the original version of Query 28 was changed at some point to "nearly empty space".) It seems to be the general opinion among Newtonian scholars that these "Aether Queries" inserted by Newton in his old age were simply attempts "to placate critics by seeming retreats to more conventional positions". The word "seeming" is well chosen, because we find in Query 21 the comments
And so if any one should suppose that aether (like our air) may contain particles which endeavour to recede from one another (for I do not know what this aether is), and that its particles are exceedingly smaller than those of air, or even than those of light, the exceeding smallness of its particles may contribute to the greatness of the force by which those particles may recede from one another, and thereby make that medium exceedingly more rare and elastick than air, and by consequence exceedingly less able to resist the motions of projectiles, and exceedingly more able to press upon gross bodies, by endeavoring to expand itself.
Thus Newton not only continues to view light as consisting of particles, but imagines that the putative aether may also be composed of particles, between which primitive forces operate to govern their movements. It seems that the aether of these queries was a distinctly Newtonian one, and it purpose was as much to serve as a possible mechanism for gravity as for the refraction and reflection of light. It's disconcerting that Newton continued to be misled by his erroneous belief that refracted paths proceed from more dense to less dense regions, which required him to posit an aether surrounding the Sun with a density that increases with distance, so that the motion of the planets may be seen as a tendency to veer toward less dense parts of the aether.
There's a striking parallel between this set of "pro-Aether Queries" of Newton and the famous essay "Ether and the Theory of Relativity", in which Einstein tried to reconcile his view of physics with something that could be termed an ether. Of course, it turned out to be a distinctly Einsteinian ether, immaterial, and incapable of being assigned any place or state of motion.
Since I've credited Newton with suggesting the second law of thermodynamics and mass-energy equivalence, I may as well mention that he could also be regarded as the originator of the notorious "cosmological constant", which has had such a checkered history in theory of relativity. Recall that the weak/slow limit of Einstein's field equations without the cosmological term corresponds to a gravitational relation of the familiar form
(1)
but if a non-zero cosmological constant is assumed the weak/slow limit is
(2)
As it happens, Newton explored the consequences of a wide range of central force laws in the Principia, and determined that the only two forms for which spherically symmetrical masses can be treated as if all the mass was located at the central point are F = k/r2 and F = l r. (See Propositions LXXVII and LXXVIII in Book I). In addition to this distinctive spherical symmetry property (analogous to Birkhoff's theorem for general relativity), these are also the only two central force laws for which the shape of orbits in a two-body system are perfect conic sections (see Proposition X), although in the case of a force directly proportional to the distance the center of force is at the center of the conic, rather than at a focus. In the Scholium following the discussion of spherically symmetrical bodies Newton wrote
I have now explained the two principal cases of attractions; to wit, when the centripetal forces decrease as the square of the ratio of the distances, or increase in a simple ratio of the distances, causing the bodies in both cases to revolve in conic sections, and composing spherical bodies whose centripetal forces observe the same law of increase or decrease in the recess from the center as the forces from the particles themselves do; which is very remarkable.
Considering that Newton referred to these two special cases as the two principal cases of "attraction", it's not too much of a stretch to say that the full general law of attraction (or gravitation) developed in the Principia was actually (2) rather than (1), and it was only in Book III (The System of the World), in which the laws are fit to actual observed phenomena, that he concludes there is no (discernable) evidence for the direct term. The situation is essentially the same today, i.e., on a purely formal mathematical basis the cosmological term seems to "fit", at least up to a point, but the empirical justification for it remain unclear. If l is non-zero, it must be quite small, at least in the current epoch. So I think it can be said with some justification that Newton actually originated the cosmological term in theoretical investigations of gravity.
As an example of how seriously Newton took these "non-physical" possibilities, he noted that with an inverse-square law the introduction of a third body generally destroys perfect ellipticity of the orbits, causing the ellipses to precess, whereas in Proposition LXIV he shows that with a pure direct force law F = lr this is not the case. In other words, the orbits remain perfectly elliptical even with three or more gravitating bodies, although the presence of more bodies increases the velocities and decreases the periods of the orbits.
These serious considerations show that Newton wasn't simply trying to fit data to a model. He was interested in the same aspect of science that Einstein said interested him the most, namely, "whether God had any choice in how he created the world". This may be a somewhat melodramatic way of expressing it, but the basic idea is clear. It isn't enough to discern that objects appear to obey an inverse square law of attraction; Newton wanted to understand what was special about the inverse square, and why nature chose that form rather than some other. Socrates alluded to this same wish in Phaedo:
If then one wished to know the cause of each thing, why it comes to be or perishes or exists, one had to find out what was the best way for it to be, or to be acted upon, or to act.
Although this attitude may strike us as silly, it seems undeniable that it's been an animating factor in the minds of some of the greatest scientists - the urge to comprehend not just what is, but why it must be so.