The question is about the function f(x) so that f(f(x))=exp (x)-1.
The question is open ended and it was discussed quite recently in the comment thread in Aaronson's blog here http://scottaaronson.com/blog/?p=263
The growth rate of the function f (as x goes to infinity) is larger than linear (linear means O(x)), polynomial (meaning exp (O(log x))), quasi-polynomial (meaning exp(exp O(log log x))) quasi-quasi-polynomial etc. On the other hand the function f is subexponential (even in the CS sense f(x)=exp (o(x))), subsubexponential (f(x)=exp exp (o(log x))) subsubsub exponential and so on.
What can be said about f(x) and about other functions with such an intermediate growth behavior? Can such an intermediate growth behavior be represented by analytic functions? Is this function f(x) or other functions with such an intermediate growth relevant to any interesting mathematics? (It appears that quite a few interesting mathematicians and other scientists thought about this function/growth-rate.)