Riemann surfaces of second kind and effective finiteness theorems

The Geometric Shafarevich Conjecture and the Theorem of de Franchis state the finiteness of the number of certain holomorphic objects on closed or punctured Riemann surfaces. The analog for Riemann surfaces of second kind is an estimate of the number of irreducible holomorphic objects up to homotopy (or isotopy, respectively). This analog addresses the problem of the restricted validity of Gromov's Oka principle.
We will discuss effective upper bounds (and in some cases also lower bounds) for the number of irreducible holomorphic mappings up to homotopy from any finite open Riemann surface (maybe, of second kind) to the twice punctured complex plane.