Victor Miller: This paper ( http://www.emis.de/journals/HOA/IJMMS/Volume12_2/245.pdf ) looks like something good to read.
--WDS: Thanks, that was of some interest. He does two things. First he describes how any finite group G can be turned into a natural iteration on |G|-dimensional vectors that is symmetric under G. In the case where the vector elements are constant on a subgroup, and on its complement, this is really only a 2-dimensional iteration. He attempts to argue the limit of this 2D iteration has a nice log asymptote only in the Gauss AGM case (and another derived-from-Gauss case) and in all other cases has a nastier oscillatory*log asymptote. Second he unearths century-old work by Meissel on a certain 3D iteration. I point out: Meisel's iteration he discusses, can really be viewed as the k=2 special case (and Gauss's AGM is the k=1 case) of the following infinite family of mean-iterations. Let A=(A0,A1,A2,...,Ak) be a (k+1)-dimensional vector, which we shall iterate. Consider the polynomial(x) = (A0+x) * (A1+x) * (A2+x) *... *(Ak+x). which is monic of degree k+1. Let its coefficient of x^j be called Bj for j=0,1,...,k. The iteration is to replace Aj for all j simultaneously by Aj[new] = ( Bj / binomial(k+1, j) )^(1/(k+1-j)) Thus A0[new] = geometric mean, Ak[new] = arithmetic mean and Aj[new] for 0<j<k are other kinds of means. This seems a fairly natural iteration when you look at it in this way. Meissel and this paper argue rather incompletely convincingly that when k=2 the limit function has a certain irrational-power-of-logarithm asymptote. I guess this would offer the fastest known way to compute that particular power of log(x) from x... if there were any reason anybody wanted to do that. Being 3D, this iteration may offer more riches than that because there are a lot more kinds of asymptotes one can consider besides Meissel's. Warren D. Smith