The elementary functions of a complex variable $z$ according to Liouville and Ritt are those functions built up from the rational functions of $z$ by exponentiation, taking logarithms, and algebraic operations. That means, the elementary functions are those functions built up by applying only $\exp$, $\ln$ and/or unary or multiary algebraic functions.
J. F. Ritt proved in [Ritt 1925] that if a bijective elementary function $f$ has an elementary inverse, then $f=\psi_n\circ\ ...\ \circ \psi_1$, where each $\psi_i$ is either an algebraic function of one variable or else $\exp$ or $ln$. R. H. Risch proved this result in [Risch 1979], in a sharpened form, from his structure theorem of elementary functions given there.
How can Ritt's theorem on elementary invertible bijective elementary functions be extended to further classes of functions?
Ritt's theorem is important for inverting functions in closed form and for solving euqations in closed form. The theorem means, an elementary function for which only function expressions which also contain multiary algebraic functions exist don't have an elementary inverse.
Risch's proof shows when a composition $f^{-1}\circ f$ is algebraically dependent over the differential field of $f$. How can Risch's structure theorem on algebraic dependency of elementary functions and Ritt's theorem deduced from that be extended to further classes of functions?
There are a number of further classes of functions representable by a differential field, e.g. subfields of the elementary functions, the Liouvillian functions, subfields of the Liouvillian fields, and functions that are generated by (generalized) composition of algebraic functions together with some (e.g. unary univalued) special functions. See e.g. [Epstein, Caviness 1979] and [Singer/Saunders/Caviness 1985].
My guess is that a bijective (generalized) composition $f$ of algebraic functions together with some differentiable transcendental functions for which only function expressions which also contain multiary algebraic functions exist don't have an inverse in the differential field generated by the functions that build the function expression of $f$.
At least the defining equations of the inverse $f^{-1}$ of $f$ with $y=f^{-1}(z)$ $\ \ $ $f^{-1}(f(y))=z$ and $f(f^{-1}(y))=z$ cannot be solved for $y$ by only transforming the equations only by inverting the operations (functions) contained in this equations. But it has to be proved if there are other functions $f^{-1}$ contained in the field generated by the functions that build the function expression of $f$.
[Epstein, Caviness 1979] Epstein, Caviness: A structure theorem for the elementary functions and its application to the identity problem. Int. J. Comp. Inf. Sci. 8 (1979) (1) 9-37