What is the expiation for one who broke an oath made to Allah and upon His Book not to do something, then did it, then repented and regretted it, and then did it again?

If you swore to abandon something permissible and then did it, there is no sin upon you. Doing it is preferable if it is better for you. However, if it was forbidden, you have sinned by doing it and repentance becomes obligatory upon you. If it was disliked, then doing it and breaking the oath in that case is disliked. In all cases, as long as you have broken the oath willingly and knowingly, an expiation for breaking an oath (kaffarat yamin) is incumbent upon you. This is to feed ten needy persons (approximately 750 grams for each needy person, though 1.5 kilograms is safer), or clothe them, or free a slave. If you are unable to do that, then you must fast for three days, and it is safer for them to be consecutive, based on the verse: "Its expiation is the feeding of ten poor people from the average of that which you feed your [own] families or clothing them or the freeing of a slave. But whoever finds not [the means to do so] - then a fast of three days. That is the expiation for your oaths when you have sworn" (Al-Ma'idah: 89).