"I Put a Spell on You" is commonly attributed to CCR since they covered it on their debut album, but it's actually a Screamin' Jay Hawkins original.
The now infamous Billy Ray Cyrus debut single, "Achey Brakey Heart", was actually a cover of The Marcy Brothers' "Don't Tell My Heart". They were released within a year of each other but the Cyrus cover exploded in popularity in Australia (yes, Australia), and the rest is history.
Cyndi Lauper's biggest hit, "Girls Just Wanna Have Fun", was a Robert Hazard cover.
Elvis Presley had a really bad habit of ripping off lesser known (black) blues artists. As in, of the 24 songs off his first 2 albums, 20 of them were covers, bad.
There's so many more I can think of off the top of my head, like Johny Cash's cover of NIN's "Hurt" getting Trent Reznor to say it "wasn't my song anymore" after seeing the video, but I think that'll do for now.