Surely one of the weirdest is John Jacob Niles' "Black is the Color." I well remember it put my mother's teeth on edge. Must have been a good few other people who couldn't stand how unresolved that one was.
Personally I always liked unresolved folksongs. We folkies don' need no steenkin' tonic note!
But I must say some of the popular orchestra song flourish endings that end up hanging like a corpse on a randomly selected note for a novelty effect give me the jimjams once in a while. Bob