Not so much an impossible request, just a difficult patron. A woman complained to my coworker that the R rated movie she checked out turned out to have material in it that was inappropriate for her young daughter. It's not that we checked the movie out to the daughter without the mother's knowledge - though we would've done that if she'd brought it up - but that the movie the mother willingly checked out, which was clearly labeled as rated R and which had the usual description of why it received that rating, had content inappropriate for children. My coworker explained, as politely as possible, that that's why it was rated R.
Same coworker, same day, had a guy come in to explain that the reason he couldn't return two books we were charging him for was that he'd been in jail. Okay, we actually get that a lot. Now that he was out, though, he still couldn't return the books because they were at his grandmother's house and she had a restraining order against him for the same unnamed offense that had sent him to jail. He was sure he'd get it all cleared up in a jiffy, but in the meantime could we call his grandmother and ask her to return the books for him, since he wasn't allowed to contact her? No, we could not.
Same coworker, gods bless her, a few days ago helped a patron make an appointment for a help session I run to learn how to use our Library2Go service to download library e-books and audiobooks on her new Kindle. That part went fine, but then the patron asked to be shown the area where we keep the e-books. My coworker asked for clarification and the conversation went around and around for several minutes, but the gist of it was the patron essentially wanted to see the e-book shelves, the physical place where we kept all the e-books for the entire state. As a gentleman on a forum I used to frequent would say, that's not even wrong.