For example, using Bring radicals or elliptic functions to solve quintic equations. Wikipedia says that similar methods can be used for higher degree polynomials, but I'm struggling on finding resources on the general theory of using special functions in addition to radicals to solve polynomial equations. I tried looking up "hyperradicals", but this name seems to be used for other purposes. Is there a name for this general theory? Has the theory I am speaking of even been developed much at all?
The theory of elliptic functions in general is sufficient to solve polynomials of any order, as is shown in Jordan's 1870 Treatise on Substitutions and Algebraic Equations. This is discussed here, for example.