@Derek: Sorry -- but how does an element of $A_7$ of order 8 look like?

Isn't the better question the other way round: "Is there a characterization of groups in which all normal subgroups are endomorphism kernels"?

@Rodrigo: What is the motivation for the particular choice of the function $f$? - I suppose there are many similar functions involving primes and divisors whose behavior under iteration is hard to describe. So what makes your function of special interest?

@Timothy: Good answer! -- I will wait some time whether someone can still tell some more, and if not, I will accept your answer.

@Emil: Indeed Nesterenko proved 'only' algebraic independence of $\pi$, $e^\pi$ and $\Gamma(\frac{1}{4})$. -- Thank you very much for pointing this out! Also, interesting that transcendence or even only irrationality of $e+\pi$ and $e\pi$ still has not been settled.

@Ricky: Obviously one can ask the question for other transcendental numbers as well - but I think I am maybe not the only one who regards π as a number of particular interest. So far I don't know examples of transcendental numbers x not defined in terms of logarithms such that ax=b for some integers a,b>1.

@Derek: I am not sure I understand your question right - depending on $m$, $n$ and $k$, the degree $d$ may of course be smaller than $\max(m,n,k)$. I performed a brief computation and found that for $d = 3, 4, 5, 6, 7, 8 and 10$ your setting permits precisely the same triples (m,n,k), while for $d = 9$, in addition (2,5,20), (2,20,5), (5,5,10), (5,5,12), (5,10,5), (5,12,5) and (5,20,2) occur (this means e.g. that there are permutations $x$, $y$ and $z$ of orders 5, 5 and 12 with sign +1 in $S_9$, but not such that $xy=z$, etc.).