What is incumbent upon someone who vowed to pray one hundred rak'ahs whenever they committed a specific sin, then became unable to fulfill this vow after the sin was repeated many times, the number of which they cannot enumerate? And is a single expiation sufficient for them, or multiple expiations?

This vow is called a vow of obstinacy and anger (Nathr al-Lajaj wal-Ghadab), and it is treated like an oath. The one who makes such a vow has the choice between fulfilling it or offering an expiation for an oath (kaffarat yamin). This is what 'Umar, Ibn 'Abbas, 'Aishah, and a group of the Tabi'in and jurists have stated.

As for Abu Hanifa and Malik, they hold that fulfilling the vow is obligatory.

The correct opinion, however, is the first one, due to the saying of the Prophet (peace be upon him): "There is no vow made in anger, and its expiation is the expiation for an oath."

Therefore, one must abstain from the sin and repent. If one returns to it, the one who made the vow has the choice between fulfilling what he vowed or offering an expiation for an oath. A single expiation for an oath suffices for all the times he committed the sin and did not expiate for it.