Generalization of Liouville's theorem

Liouville's theorem deals with an elementary differential extension, composition is considered there. But your problem contains the additional operation inversion.

Therefore your problem is not a generalization of Liouville's theorem but a different task.

$E_{i+1}\setminus E_{i}$ contains the non-elementary antiderivatives of the functions from $E_{i}$ and the non-elementary inverses of the functions from $E_{i}$.

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 $E_{i}$ that have a non-elementary inverse.

Your $E_{i}$ 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.