Brandy, there is a one-night limit, but the place is never full, and the rule seems not to be enforced. There are nice day hike opportunities there: Cottonwood Pass, Cottonwood Lakes, Trail Peak.

It's about 30 minutes down to Lone Pine, so you could head down for a big dinner at a restaurant, maybe a shower at the Whitney Portal Hostel, then back up to sleep at altitude.