What is incumbent upon one who swore to abstain from a forbidden act, then broke the oath many times without expiation, then swore again concerning the same act, such that a new expiation becomes due upon him, other than the first?

It is obligatory to repent to Allah for committing a forbidden act. If you break the second oath you swore as a self-imposed punishment, then according to the majority of scholars, you are liable for two expiations: the first for breaking the oath by doing what you swore to abandon, and the second for failing to offer expiation. The expiation is not multiplied by repeated breaking of the oath because you did not use the phrase "كلما" (every time/whenever). However, if you did not break that second oath, then you are only liable for one expiation, which is the expiation for the first broken oath.