It feels bad talking about a space without knowing a single function on it, hah?
So what is a function on the moduli space of curves, from the geometric point of view?
From the functorial point of view, it should be invariants of family of curves, which is natural w.r.t. pullback of families. The j-invariant maybe one for M_1. But does anybody have a concrete example for higher genus?
I do have two guesses of sources of functions:
There are some geometrically defined divisors, maybe take one, and then pick two sections of the line bundle it defines, and take their quotient?
Maybe there are some "natural" differential form, whose integral over the whole curve is a function on M_g?