The question is about the function f(x)$f(x)$ so that f(f(x))=exp (x)-1$f(f(x))=\exp (x)-1$.
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$f$ (as x$x$ goes to infinity) is larger than linear (linear means O(x)$O(x)$), polynomial (meaning exp (O(log x)))$\exp (O(\log x)))$, quasi-polynomial (meaning exp(exp O(log log x)))$\exp(\exp O(\log \log x))) $ quasi-quasi-polynomial etc. On the other hand the function f$f$ is subexponential (even in the CS sense f(x)=exp (o(x))$f(x)=\exp (o(x))$ ), subsubexponential (f(x)=exp exp (o(log x))$f(x)=\exp \exp (o(\log x))$) subsubsub exponential and so on.
What can be said about f(x)$f(x)$ 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)$f(x)$ 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.)
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