Implicit Means

Consider the three most common "means" of two variables x,y

 A(x,y) = (x+y)/2      G(x,y) = sqrt(xy)    H(x,y) = 2/((1/x)+(1/y))

We can construct another binary "mean" function by combining these 
three common means as follows

                       _____
            x ________|     |
                  ____|  A  |__
                 |    |_____|  |   _____
                 |             |__|     |
        _________|              __|  H  |----*----- z
       |         |     _____   |  |_____|    |
       |         |____|     |__|             |
       |    y ________|  G  |                |
       |              |_____|                |
       |                                     |
       |_____________________________________|

Let this function be denoted by AGH(x,y).  By permuting A, G and H we 
can construct six such functions, but since the first two components 
are symmetrical we have only three distinct functions, which we will 
call AGH, AHG, and GHA.  The function z = AGH(x,y) implies that z (if 
not zero) satisfies the cubic

           z^3  +  (2x) z^2  +  (x^2) z  -  (4x^2 y)  =  0

which, in the special case x=y, factors as

                 (z-x) (z^2 + 3xz + 4x^2)  =  0

Therefore, if z = AGH(x,x) then either z=x (as one would expect) or 
else z equals one of the two quadratic roots

                         -3 +- sqrt(-7)
                         --------------- x
                                2

If we consider the HGA mean we find z = HGA(x,x) implies that either 
z equals x or else z equals one of the values

                         -3 +- sqrt(-7)
                         --------------- x
                                8

Interestingly, for the third mean, AHG, we find that AHG(x,y) = G(x,y).

Anyway, given the three original two-input functions, we have 
constructed another set of three two-input functions.  These can 
be combined in the same three distinct ways to produce yet another 
set of three functions, and so on.  At each level (above the lowest) 
the three functions have precisely the same structure in terms of 
the three lower-level functions. 

QUESTION:  What is the limit of these functions at the nth level, 
           as n goes to infinity?