There is a South American culture described in Bruno Nettl's book on ethnomusicology that took this to the ultimate extreme. Everybody in the culture had their own personal song, which they'd composed themselves. Nobody else would ever sing it and you would never sing more than the one.

Nettl doesn't say, but I suspect they took up headhunting and cannibalism so as to hear some bloodcurdling screams as a change from the weekly singaround.