The song can have one instrument or one voice, but NOT both (in other words, either a solo instrumental, or a solo acapella voice), and must be performable without multi-tracking. I imagine the classical charts contain such recordings, like piano performances, but what the US Billboard Hot 100?
My candidate would be "Mercedes Benz" from Janis Joplin. Problem with the billboard is that it was the B side of the "Cry Baby" single (placed #42) and the last recorded track (she died few days later) on the LP "Pearl" (placed #1) so it's hard to give it a rating of its own. It's been one of the more iconic performances and a lot of other singers made cover versions, though.