Consider the equation
$$ f(f(x)) = \exp(\exp(x)) $$
Valid for all real $x$, $f(x) \neq \exp(x)$ (not identically equal everywhere), $f(x)$ is analytic and
$$ f(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + ... $$
Where $a_1,a_2,a_3,\cdots > 0 $.
Find solutions for $f(x)$.
I wonder about the behaviour of such an $f(z)$ on the complex plane.
Where does it satisfy the main equation NOT on the real axis? What about singularities? How does it grow in the imaginary direction? Where are its zeros etc.
But these are not the main question.