My main question is how to solve this functional equation: $f(f(x)) = e^x$.
The context within which this came up was when I was attempting to extend the notation of the inverse function ($f^{-1}$) to include something like $f^{0.5}$, where composition of the functions adds the superscript numbers.
In this extension $f^{1} = f$ and $f^0$ is the identity function, so $(f^{0.5} \circ f^{0.5}) = f$. So as an extra question, for which classes of real-valued functions does there exist such an $f^{0.5}$? It's fairly easy to see that for every polynomial function with a single term, represented by $f(x) = ax^n$, $f^{0.5}(x) = \sqrt[\sqrt n + 1]{a}x^{\sqrt n}$. I'm not sure how to proceed from there though.