Is there any analogue of the method for expressing roots of polynomials of degree 5 with elliptic and η-functions that generalizes to polynomials of degrees n>5?

More specifically: does there exist any ''reasonable'' sequence of ''reasonable'' finite sets of ''reasonable'' special functions such that for arbitrary polynomial of degree d its roots are expressible by a unique term made out of coefficients of polynomial and functions from the d-th set?

By ''reasonable'' special functions I mean ones that has been investigated independently or have been already coined and are decently understood or are solutions to reasonable ODEs.

Are there some general results on restrictions to such sequence of special functions? Existence of integer solutions to polynomial equations is an undecidable problem. So if we replace ''reasonable'' by ''recursive'' and define ''reasonable special function'' as one defined by a power series with rational coefficients $(a_n)$ computably depending on $n$, then I imagine the answer might be 'no'. But no obvious proof comes to my mind that such a family may not exist.