How competitive is a Teton Crest Trail permit and what are the group rules?

Very competitive, and the calendar is unforgiving: Grand Teton backcountry advance reservations open in early January (January 10), and prime August dates in the popular camping zones go almost immediately. The permit is a reservation through Recreation.gov rather than a lottery, so speed and flexibility on the opening day matter more than luck.

Group rules are stricter than most western trails: maximum group size is 6 for backcountry camping. Bear canisters are required (this is active grizzly habitat the whole way), and campfires are prohibited above 6,800 feet, which in practice means the entire route.

One structural quirk works in your favor. Alaska Basin, the wildflower basin around mile 15, sits in the Jedediah Smith Wilderness outside the park boundary and requires no permit at all. Itineraries built around an Alaska Basin night need fewer in-park camp reservations, which makes a patchy reservation calendar workable. Plenty of successful Crest trips have been assembled around exactly that trick.

If January goes badly, cancellations trickle back through spring and summer as groups drop out, and watching for those returned dates is the realistic second chance. The route itself is about 40 miles with 9,000 feet of gain over 3 to 5 days, so most parties need two or three camp nights inside the park at most.