What is the expiation for someone who vowed to pray two rak'ahs with every (obligatory) prayer as thanks to Allah, then was unable to adhere to that, and does not know the extent of the prayers he missed?

Whoever vows obedience to Allah, it is obligatory for them to fulfill their vow, based on the saying of the Prophet, peace and blessings be upon him: "Whoever vows to obey Allah, let him obey Him."

So, if you vowed to pray two rak'ahs with every obligatory prayer, it is incumbent upon you to fulfill this vow. Whatever you missed of it is a debt that must be made up. Furthermore, according to the Hanbalis, an expiation is added to the making up due to not performing the vowed act as specified.