What is required of one who vowed to fast and recite Quran in order to refrain from a sin, but then committed the sin multiple times, to the extent that he does not know how many times he is obligated?

The "oath of contention" is what this type of vow is known as. If the one making the vow says, "Whenever I do such-and-such, I must fast three days and recite the Quran," or intends that meaning, then he owes an expiation for an oath (kaffarat yamin) for each time the action is repeated. The expiation for an oath is feeding ten poor persons, or clothing them, or emancipating a slave. If he cannot find any of these, then he must fast for three days. If he does not know the number of times he broke the vow, he should make an effort and offer expiation until he is reasonably certain that he has atoned for all of them. However, if he says, "If I do such-and-such, I must fast three days and recite the Quran," and does not intend the vow to be repeated with the repetition of the action, then he owes only one expiation for an oath. One must be careful not to commit disobedience to Allah after vowing to abandon it; rather, he should hasten to repent and increase good deeds.