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Let $f$ be a real analytic (on at least $|x|<2$) and real solution of the functional equation $f(0) = 1,f(x+1) = \exp(f(x))$. For the existence of such $f$, see here. Then $$ f(x) = \sum_n a_n x^n ;...

While talking about tetration with my friend the following idea (re)occured. $$f(f(z)) = z ,\quad f(\exp(z)) = \exp(f(z)) \tag{A}\label{A}$$ or variations of it like the weaker $$f(f(f(f(z)))) = z ,\...