  The nef cone volume appeared first in work of Peyre in a number-theoretic
context on Del Pezzo surfaces, and it was studied by Derenthal and co-authors
in a series of papers. The idea was subsequently extended to also measure the
Zariski chambers of Del Pezzo surfaces. We start in this paper to explore the
possibility to use this attractive concept to effectively measure the size of
the nef cone on algebraic surfaces in general. This provides an interesting way
of measuring in how big a space an ample line bundle can be moved without
destroying its positivity. We give here complete results for simple abelian
surfaces that admit a principal polarization and for products of elliptic
curves.
