Blinded by the Light/ Mandred Mann
Almost the entire album of Nirvana Unplugged in New York
Rivers of Babylon/ Sublime
Total Eclipse of the Heart/ Bonnie Tyler
Love Hurts/ Nazareth
Turn, turn, turn/ The Byrds
Twist and Shout/ The Beatles
Almost every cover song Cash or GN'R did I like better than the original
I also like how Joe Cocker got stoned with help from his friends, and then covered a song about it.
I'm going to assume that the only reason A Perfect Circle's eMOTIVe record isn't mentioned here is because it was such an obvious choice that it was overlooked.
That's why I haven't said anything until now. I often forget these are covers.