Music: Great covers that were never recorded

I find this to be a pretty exciting thought experiment / dream away kind of excercise. What other bands'/musician's songs should sound amazing if performed by someone else?

I'll start the thread with:

Pixies - (I can't get no) Satisfaction [The Rolling Stones] One key strength to the Pixies' music was always the dual vocal genius of Francis/Deal. That this song fits amazingly well for a duet was proven by Björk and PJ Harvey over a decade ago - but with the added dimension of male/female dynamics, I think this song could reach stratospheric heights in terms of brilliance. Adding to that, of course, a pressumably awesome verse/chorus dynamic, as with most songs that the Pixies recorded.

Johnny Cash - Sabotage [The Beastie Boys] I'm not really sure of why and how, but I think this could have sounded wonderful. In his later years, of course. But then again, if it never ocurred to Rick Rubin - maybe there's a reason to that.

Dusty Springfield - Professional Widow [Tori Amos] One all-time great singer interpreting another's song. Nothing too funky about that. But just sit back in silence and imagine what it'd sound like. Imagine Dusty uttering the words "star ****er, just like my daddy", over and over again. It would be pure genius.

More to follow from me, surely. But I expect this to be a topic which success isn't too dependent on the OP anyway.