What is the minimum mass required so that objects become spherical due to its own gravity?

This question is more complicated than it seems like it should be!

There is no threshold mass or density beyond which an object becomes perfectly spherical; even supermassive stars are slightly oblong. The only exception is black holes, which are perfectly round up until you reach the quantum level. If we want a simple answer, most guesses are somewhere around $frac{1}{10000}$ the mass of earth, or $6cdot10^{20}$ kg, but that is very approximate and depends on the composition of the object.