Seinfeld has talked about how actor Wayne Knight who played Newman on his long-running show CONSTANTLY has to cope with people saying "hello Newman" to him, based on a catchphrase from the show. Similarly, the actor who played Biff from back to the future has developed a frequently asked questions on his business card.
Characters that have one movie or show or public moment and get pegged as those people may not exactly have it rough, but I do empathize. The whole fun of being famous seems to be recognition and appreciation by a wide audience. Not dull repetition.
Incidentally, Wayne Knight was a very enjoyable villain on Toy Story 2.