TALKEETNA, Alaska—The mayor of this tiny village has been shot, fallen into a restaurant fryer, jumped off a moving truck and been mauled by a dog. Now the burning question around these parts is: Has Mayor Stubbs used up his nine lives?

Stubbs is a cat—but that didn't stop residents of this unincorporated burg of 876 from naming him their mayor 16 years ago. "It's an honorary position we gave him, and it just stuck," says Lauri Stec, general manager of Nagley's General Store, where Stubbs was adopted by the management...