What is the expiation for someone who vowed not to do something specific, but then did it more than once? And is he required to offer an expiation for each time, or just one expiation?

This type of vow is called: a "vow of contention" (nadhr al-lajaj). The preponderant opinion regarding it is that an expiation for a broken oath (kaffarat yamin) is required if the one who made the vow does not fulfill it. As for the repetition of the vow with the repetition of the act, it depends on whether there is an indication of that in the wording or the intention. If such an indication exists, the vow is repeated; otherwise, fulfilling the vow once is sufficient.