# Generalization of Liouville’s theorem

As proposed in this answer, I wonder if the answer to following question is known.

Let $E = E_0$ be the set of elementary functions. For each $i > 0$, inductively define $E_i$ to be the closure of the set of functions whose derivative lies in $E_{i-1}$ with respect to multiplication, inversion, and composition. Does there exist an integer $n$ such that $E_n = E_{n+1}$?

This seems like such a natural generalization of Liouville’s theorem, it has to have been asked before. After a couple of quick internet searches, I can’t seem to find anything.

$$E_{i+1}\setminus E_{i}E_{i+1}\setminus E_{i}$$ contains the non-elementary antiderivatives of the functions from $$E_{i}E_{i}$$ and the non-elementary inverses of the functions from $$E_{i}E_{i}$$.
With a generalization of the theorem of Ritt of Ritt, J. F.: Elementary functions and their inverses. Trans. Amer. Math. Soc. 27 (1925) (1) 68-90 which I hope to prove, one could show there are elementary functions in each of your $$E_{i}E_{i}$$ that have a non-elementary inverse.
Your $$E_{i}E_{i}$$ are therefore no differential fields and you cannot apply Liouville’s theorem. Therefore your problem cannot be solved by the Liouville theory treated in the literature.