Back on Scott Aaronson's blog, I gave an argument that $e^z+z-1$ should have an analytic compositional square root. The important difference between this function and $e^z-1$ was that the fixed point at $0$ has derivative $>1$, not $=1$. This should warn us that arguments based on the growth rate near infinity are inadequate. (Or else it should point out that my argument was broken!)
See comments below, my argument may have been broken. But, if so, I want to figure out why!