Sonny & Cher - 'I Got You Babe' (1987)
In November of 1987, Sonny Bono was running for mayor of Palm Springs and Cher was promoting her upcoming movie Moonstruck. The divorced couple went on Letterman together and were coaxed into playing "I Got You, Babe" with Paul Shaffer and his band; they weren't exactly the best of friends at this point, but both of them knew it would make for unforgettable television. Had YouTube existed back then, this would have gone insanely viral the next morning. It wound up being the final time they ever sang together, as Bono passed away in January 1998.