Some folks are weird, and I leave it at that.
Even Vegan Death Metal exists, c.f. “Cattle Decapitation”
Though, unless ypu are a fan of fringle metal, I’d not suggest listening to their music. Even my most metal fan friend can only take it in small doses.
As for music that simply cannot be covered, I profer Eric Idle with the hit single, “Always look on the Bright Side of Life”.
Sorry, can’t help you there. We were a pretty sniffy bunch. 
The best not-a-punk story was about John Otway, who only got a record contract because the A&R man involved thought he was a punk. Clue: he’s not, he defies categorization. Nobody will ever cover Otway’s version of House Of The Rising Sun.
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I think Johnny Cash did a decent job with “Down There By The Train”.
I beg to differ on this one. I give you Brown Sabbath:
That was awesome.
Idiot Wind is the first one that pops into my mind. The uncoverability of a Dylan song is directly proportional to the amount of sneering vitriol in the original. No one sings anger like Dylan.
Metal Machine Music?
A couple friends of mine tried to do it a cappella, but petered out after a minute or so.
No one but Justin Hayward should sing the Moody Blues songs that Justin Hayward wrote.
For example: Nights in White Satin
Tuesday Afternoon
Question
Gypsy
etc. etc.
They weren’t the originators of these songs, but no one should perform them, but them:
Tiptoe Through the Tulips
Shine On Harvest Moon
I’m surprised no one has mentioned the Beatles in this thread. I generally don’t care for Beatles covers (though I’ve heard a few that were OK), but one jumped to mind as I read through this: “I Want to Hold Your Hand”.
My thought is that, while this is not such a personal song, it is the iconic Beatles song for Americans of a certain age. And no one else could re-create the impact that song made on those of us of a certain age.
It hasn’t really aged that well, but it means more than just the song itself.
Nobody should cover anything Chris Cornell sang, because it will suck in comparison.
Bravo!
Weird Al Yankovic actually did a parody. It wasn’t a parody of the actual song, but it’s a style parody of Dylan and the video is a parody.
Weird Al Yankovic did a parody. Not a cover of the exact song. It was a style parody and the video was a parody of the original video.
And this one. Sorry Barry, Robin and Maurice but he nailed it and he owns it.
I was in a Thai restaurant in Orlando when I heard an upbeat soft jazz version of “Black Hole Sun”. No-one was killed in the aftermath but it was a close-run thing.
I can’t think of too many many serious attempts at covers of “The Who”. Can’t think of any that would work. Also, for someone who wrote many songs for others there are still iconic songs by Prince that are best left alone. “Purple Rain” probably high on that list.
An album of Who covers reached #2 on the charts. I’d venture to say more people have heard Elton John’s version of “Pinball Wizard” than The Who’s.
Christian Death Metal doesn’t make much of a contradiction, but Christian Black Metal would.
In a hippy coffee house, I once heard a geeky white guy perform “Purple Rain” with an acoustic guitar, and it STILL sounded good. Sinead O’connor and the Bangles both had hits with Prince songs. He wrote songs for everybody to perform