What is the expiation (kaffara) obligatory upon one who vowed to abandon masturbation, then broke his vow three times, and is expiation required for each time he broke it?

You must strive to achieve sincere repentance from the bad habit. If you fall into it after resolving to abandon it, repent to Allah again and strive until you become steadfast in your repentance. Do not despair of repentance, even if the sin is repeated. Imam Al-Nawawi said: "If a sin is repeated a hundred times, or a thousand times, or more, and one repents each time, their repentance is accepted, and their sins are forgiven."

As for vowing to abandon that habit and the obligation to fulfill it, abandoning what is forbidden is a religious duty. A vow to abandon what is forbidden emphasizes the obligation to abandon it. This type of vow is called "Nathr al-Lajaj" (a vow made in anger or a state of agitation), and the one who makes it has the choice between expiation and fulfilling the vow.

If the wording of the vow does not imply repetition, such as "I must fast three days if I engage in masturbation," and you engage in it three times, then you are only obligated to offer one expiation for an oath (kaffarat yameen) or fast three days.

However, if the wording of the vow implies repetition, such as "I must fast three days whenever I engage in masturbation," then the expiation for an oath is repeated according to the number of times you engage in the habit after the vow, or you fast nine days if you engage in it three times.

The expiation for an oath is feeding ten poor people, or clothing them, or freeing a slave. If you cannot find any of these, then it is fasting for three days.