What does everyone do about large members involvement?

Our team (663) has been involved with FIRST for four years. We're a small team, but we still struggle with trying to provide an opportunity for everyone to contribute. Whenever I go to a regional competition, and see very large high school teams, I'm left wondering what everyone on the team does.

Our robotics club isn't directly connected to any curriculum at the school, so the kids who join the club all want to get their hands dirty playing with robots. A limited number actually get to work directly with the robot construction, so we have a pretty high turnover rate as the less-involved kids drop out. I'd love to have 30-40 kids on the team, but what would they all do?