You are right, it is hard to make a hard rule that says you are no longer a beginner, and I don't think that there should be, because, as you say, there are many different things that you can be a beginner at. However, I think that someone who comes in and wins 5 beginner jams in a row is probably not doing it to improve their skills, but rather to boost their ego, and beat up on new people. I am not saying that there should be rules to prohibit people from competing multiple times, but rather there should not be rules to encourage people to come in and repeatedly beat new developers. Once you feel you are no longer a beginner, you should step aside, and let fresh blood come in and take a shot at winning. But why would you if there is a rule that says you get extra recognition when you win consecutively?
I would never dream of winning a game-jam anyway, so I'm not really worried about anybody winning 100 times in a row. For me its about ppl playing my game and giving me feedback/leave a comment if/how they enjoyed the game I created. If I get ppl to try my game and tell me about their experience, thats a win for me.
Thats the way I personally approach game-jams, also when I go though the submissions. I'm usually not focusing on the slickest games (especially in a beginners jam) but on random games. If I feel ppl put some effort in what was produced, I make sure I leave a comment because, again, thats what I get the most out of with game-jams.