The answer to this depends a lot on variables in the steeping process. Steeping at a higher temperature will remove more caffeine. Steeping for longer will remove more caffeine. Doing either or both of these will leave less caffeine for a second or third (or more) steeping. Using whole leaves can slow down caffeine extraction, while using fannings (as in tea bags) can speed it up.
For example, some green and white teas are steeped repeatedly for very short (~30-second) intervals. The water temperature is often quite a bit below boiling as well. In that case, the second steeping can potentially contain more caffeine than the first (as I noted in a previous answer). For a longer initial steep, the caffeine extraction for that first brewing will gradually increase, though you will get diminishing returns for very long steeps (as discussed here).
Note also that these steeping conditions significantly affect perception of caffeine content and actual extraction. As described in a previous answer, white tea is often thought to have less caffeine than black tea (mentioned in the question), but that is likely due to the fact that white and green teas are typically brewed at lower temperatures. The total amount of caffeine you might extract from a white or green tea in multiple infusions may be roughly the same as black tea. And when all types of tea are brewed under similar conditions, they will generally extract similar amounts of caffeine.
Thus, there's no simple equation to describe the relationship between the caffeine content of multiple brewings. The higher the temperature and the longer the first steep, the more caffeine will be extracted initially, leaving less caffeine available for subsequent brews made with the same leaves.
If you follow a procedure as suggested in the question (5 minutes, almost boiling water), Dugan's answer of roughly 70% caffeine extraction is a pretty reasonable estimate. In general, in those conditions, I'd say a good rule of thumb is that each steeping will extract about 2/3 of the caffeine left in the leaves. (So, first steeping is roughly 67% leaving about 33% in the leaves. Second is roughly 22%, leaving about 11% in the leaves. Third is roughly 7-8%, leaving only a tiny about of caffeine left.)
But again, that only applies to that specific scenario. Various tea studies Dugan and I have linked to can give estimates for other conditions.