The one that surprised me was “Seasons In The Sun” originally being a Belgian art song.
“I Want Candy” was originally recorded by The Strangeloves (not Bow Wow Wow).
It’s My Life, No Doubt. (Talk Talk).
I just realized that I made it sound like I don’t like it anymore. I do! 
In fact, I like both versions.
I came here to mention the prolific, talented and heartbreaking work of Jacques Brel.
I’ve often ranted about ill-translated covers of his work. Rod McKuen’s adaptation of Le Moribund - a satire where a dying man ironically praises those who have helped his wife’s infidelity - into Terry Jacks’s insipid, lovelorn ‘Seasons’ drives me a bit mad.
Anyway, he is also the same singer/songwriter who gave us:
If You Go Away - Terry Jacks*
The Port of Amsterdam - David Bowie
I’m Not Afraid - Frank Sinatra
I belong to your heart - Sam Cooke
Jacky/ie - The Divine Comedy/Scott Walker
If we only have love - Barry Manilow
La chanson des vieux amants - Alison Moyet
And dozens of others. (Bonus Brel: here’s a fun, tragicomic song of his with subtitles that isn’t covered very often.)
*Another lame McKuen translation, turning a heart-wrenching plea that is so raw that his live performance is almost painful to watch, but emotionally true, into a wistful hypothetical.
Having never seen Hair, I thought Frank Mills was a song by The Lemonheads.
WHAT??? I had no idea.
Man, one thing about buying CDs is that I never read liner notes anymore.
Oh, I miss this so much. Remember opening an album, putting it on the turntable, kicking back into your beanbag and reading the liner notes (and lyrics) while you listened? Sigh.
When Sixpence None the Richer came out with “There She Goes” I never heard of the original performers, The La’s.
When Orgy did “Blue Monday” I didn’t know it was a New Order song.
“You Ain’t Nothin’ But a Hound Dog” was originally recorded by a woman whose last name was Thornton.
Also, a number of songs from “Godspell” had lyrics written by other people, including “Day by Day”, “All Good Gifts”, and “God Bless the People”.
I had always thought “Oye Como Va” was written by Santana. Until one day, I heard Tito Puente. I have both now and quite a few more of Tito’s arrangements. Thank you Carlos Santana!
Willie Mae “Big Mama” Thornton. She also was the first to record “Ball and Chain,” later a hit for Janis Joplin.
William Shatner’s “Mr. Tambourine Man” was recorded earlier by the Byrds and Dylan.
Did anyone think Shatner’s Mr Tamborine Man wasn’t a cover???
I suspect you’re being whooshed.
I found this out only a few days ago when I was listening to Pandora. “There, I’ve Said It Again” by Bobby Vinton, which is about as white-bread as you can get, had been recorded four years earlier by Sam Cooke. However, not even Cooke’s version was the original one since the song had been recorded by other artists since the 1940s.
“Without You” as sung by Nilsson and (horrors) Mariah Carey is a cover of a Badfinger song.
Sometimes I have to remind myself how much younger most of you are. I bought the semi-original, Broadway, cast recording of “Hair” many years too late, but still before many of you were born. As is the case with too many of the examples you folks list.
Well, I’m not that young. I’ve just never watched Hair or heard all the songs on the soundtrack. The main reason probably being that I don’t really like musicals. 
(psst! I gave you an out but you are letting your age show!)