Between polygons in $\mathbb C\cup\{\infty\}$ (including the "single side polygons", hemispheres, disks) the Schwartz-Christoffel mappings give arguably explicit conformal maps. For polygons with few angles those are well known special functions -hypergeometric for triangles to halfplanes and elliptic integrals for rectangles. We also have domains with piecewise quadratic boundaries to which we can write explicit transformations. For instance sectors of disks map to disks via powers -removing the origin to make this conformal- and to halfplanes via Möbius transformations.

My question is whether we can write explicit transformations between domains piecewise bounded by higher degree real algebraic curves, like cubic or quartic, and a disk. There are numerical methods but can we use special functions?

This is related to calculating periods in the sense of Kontsevich-Zagier, though my motivation is rather low-level.

EDIT: I may consider as "explicit" integrals of algebraic functions. If you can give formulas in the form of "generalized Schwarz-Christoffel integrals" but for arbitrary polynomials defining piecewise the boundary that would already be satisfying.