In this chapter we apply the concept of conformal module and extremal length to obtain quantitative statements on the limitation for Gromov’s Oka Principle. More specifically, for any finite open Riemann surface X (maybe, of second kind) we give an effective upper bound for the number of irreducible holomorphic mappings up to homotopy from X to the twice punctured complex plane, and an effective upper bound for the number of irreducible holomorphic torus bundles up to isotopy on such a Riemann surface. These statements are analogs for Riemann surfaces of second kind of the Geometric Shafarevich Conjecture and the Theorem of de Franchis, that state the finiteness of the number of certain holomorphic objects on closed or punctured Riemann surfaces.