Is tetration a transcendental function?
If so are there any papers with a proof?
I suspect that it is because I have not seen any algebraic situations where tetration is the answer and the fact that it has little active research.
For those unaware of tetration: $${}^na=\underbrace{a^{a^{a^{.^{.^{.^a}}}}}}_{n \text{ times}}$$ and assuming it's extension to $\Bbb R$ and $\Bbb C$.
Hint $\ $ Already $x^x$ grows faster than any algebraic function (which is asymptotic to $x^q$ for $\,q\in \Bbb Q).$ And they grow even faster as you tetrate $f \mapsto x^f.\,$ Hence for $\,n>1,\,$ they are transcendental, i.e. not algebraic functions.