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The recent proof of Serre's conjecture allows one to prove the finiteness of certain types of Galois representations (2-dimensional
mod $p$, odd, semisimple respresentations of the absolute Galois group of the rationals with bounded prime to $p$ Artin conductor). One may ask for  quantitative refinements of this. It seems hard to get a grip on this, although one can formulate a natural expectation.

This is in the spirit of the theorem of Hermite-Minskowski which asserts that there are only finitely many extensions of the rationals of bounded discriminant. One can ask for quantitative versions of this: a precise count is again unknown.
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