A number of mine have already been mentioned (mostly Radiohead, The Beatles) but here are a few that haven't.
Paul McCartney has said that his is one of his favorite songs of all time, he broke into tears the first time he heard it:
Anything from the Kind of Blue album
Reid Jamieson's cover of The Everly Brothers classic "Let it be Me"
http://www.reidjamieson.com/reidoradio/LetItBeMe.mp3