My library (which does still have late fees) does this. You can get 5 books at a time, initially, and you can keep them for 21 days, and you can renew by phone

My library (which does still have late fees) does this. You can get 5 books at a time, initially, and you can keep them for 21 days, and you can renew by phone once per book (so long as they're not on reserve for someone else). If you have had your card for >90 days, and have borrowed and returned >10 books, you instead are allowed 20 books at a time (no more than 5 per topic/category), and can renew twice before returning. But if you run up too many fees, you go back to the 5-book-limit.