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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!)