There's two different categories here: the songs you get tired of hearing other people do, and the songs that you're tired of performing yourself. In our Open Sing crowd, some people will remember you for one song of all the ones you've ever sang - and it inevitably becomes one you wish no-one had ever heard you sing! Even if it's a good song, people will ask you to sing that one song over and over and over...
One friend, a lovely singer with a fine repertoire, constantly gets requests for "Five Constipated Men". There was one person who would always ask my husband to sing Wade Hemsworth's "Black Fly" song - never anything else. Even though we're all (mostly) amateurs in our singing circle, it feels like the stories of Jerry Jeff Walker threatening to strangle the next person who asked him to sing "Mr. Bojangles" - nice song, but, good grief, he does so many other wonderful songs as well!
I guess that's the curse of the "One-Hit Wonder" mentality in audiences...folks with a short attention span can only remember you for just one thing.