actually, i routinely renew books at my local branch of the Dallas Public Library for WAY more than two times. I do it online and it's very convenient. You just log on with the number from your card and (as another poster mentioned) as long as there isn't a request for that book, I have renews TONS of times. I just keep track and go back in and renew as it gets closer if i didn't finish the book(s) yet...i get a lot of books about my business and the ingredients I use for the products i make...not just fiction...so i keep them a while and not many people request them.
I also do my searching for books from home (not the library computer) and request books and they send them right to my branch and all i have to do is walk in and give my card and there are my books....it's absolutely great! if you are in dallas the webpage is dallaslibrary.org. Don't put the www. at the front.
good luck. that's great that your daughter is so smart and such a reader.