That really depends on how the regeneration effect works.
If it's biological (wolverine) then regeneration happens on the cellular or sub cellular level and requires some materials and energy. If wolverine stopped eating and got hurt, he would eventually lose his ability to regenerate. Depending on how much he got hurt, this may happen before he actually starves to death. Since cells can be motivated after brain function ceases, this also explains how wolverine can heal from "instant-death" wounds (stabbing the heart, brain damage, etc)
If it's a magical effect (as in thresh's lore) then a whole other mess of problems arises. Most of these depend on what the rules of magic are in the universe and how the magic actually works. If magic must consume some energy to work, then it could be very similar to wolverine's mutation - as long as your body has energy, the magic heals it. If the magic is an outside source that doesn't require energy, then any limitations is has depend on how specific the spell is.
In this case, it sounds like the prisoner has some sort of outside magic that does not require energy but only affects parts that are still attached to him (if you rip off skin, the skin won't do anything, but the body will regrow the skin). In this case, it would be impossible to kill all his cells, since they would just reanimate. Damaging his memories may or may not be possible, depending on how the magic heals. If it regenerates the brain tissue to the exact same state, down to the neural connections, the memories should be intact.
I think there is no physical way to kill this person (aside from reducing him to his base elements in a nuclear blast, sending him to the sun or into a black hole) - you have to resort to means outside physics as we know it. Either removing his soul, stopping the magical regeneration effect (either directly or by finding any limitations it has) or using other magic to obliterate him in some manner.