Perhaps there is a simple answer, but I'm very puzzled by the following question:
Question: Does there exist a (smooth, connected) algebraic group $G$ such that the general centralizer (i.e. the centralizer in a Zariski open set) is finite?
I'm pretty sure the answer is no (maybe some extra assumptions such as reductive, algebraically closed field... must be added), but I have no clue about how to prove it.
Of course, typical groups such as $GL_n, SL_n, O_n...$ are not counterexamples.