I created a thread a few years back about the definitive versions of famous Christmas songs, and this takes that to the extreme. Are there any songs with only one *significant *version? If someone obscure make a recording in their basement and put it Youtube it wouldn’t count, but anything released by a recognized label would.
To start - Dominick the Donkey by Lou Monte
I was going to vote ¿Dónde Está Santa Claus? by Augie Rios. But ¡no! It has several. Go figure.
I wonder if there will be many.
I thought Christmas Wrapping by The Waitresses might make it, but there are lots of cover versions there too.
How about “The Twelve Days of Christmas,” Bob and Doug MacKenzie lyrics? It may end up being parodies of more famous songs that haven’t been covered.
I was going to suggest the Chipmunk song (Christmas Don’t Be Late) but nope; a Google search turns up other versions.
True – what about the several Bob Rivers “Twisted Christmas” albums? I’ve never heard anyone cover the Black Sabbath-inspired “I Am Santa Claus!” for instance
(Although the folks at WROR here in Boston did a cover of “Walkin’ Round in Women’s Underwear”)
Google doesn’t turn up any covers of Spinal Tap’s “Christmas With The Devil” or Pansy Division’s “Homo Christmas”.
Appallingly, there are many, many covers of “Wonderful Christmastime”.
Christmas At Ground Zero by Weird Al is a one-of-a-kind.
I wonder if our list will be mainly novelty Christmas songs?
Amazon lists at least a dozen covers but nothing on a major label. Most of them seem to be from an outfit called Blue Paradise Records.
It apparently took 50 years before someone started making covers of Santa Claus got stuck in my chimney, but now there are at least four …
We Wish You The Merriest, by Frank & Bing seems to fit the bill.
It’s bewildering how many otherwise legitimate musicians make Christmas albums every year. Because of this, I’d say the vast majority of Christmas songs have no covers. Ryan Adams has a Christmas song, did anybody know that? I doubt anyone has covered it. Nick Lowe, NRBQ, Bob Dylan, Old 97s, and thousands more musicians I could name off the top of my head but won’t, have released Christmas albums. If any of their Christmas songs aren’t covers already, they’re probably one-offs nobody else has covered. There are only a few rare “Little Saint Nicks” and “Grandma got run over by a reindeers” that have been added to the canon since the 1950s.
Even Bowie and Bing’s Peace on Earth/Little Drummer Boy has a cover that charted. I’m amazed.
Pretty sure nobody’s bothered to cover any of Meco’s Star Wars Christmas Album. The originals are bad enough!
All tracks should be on YouTube, if anyone’s interested.
“Christmas at Kmart” by Root Boy Slim and the Sex Change Band doesn’t seem to have a cover.
I discovered Martin Mull’s “Santa Doesn’t Cop Out on Dope” has been covered, but his "Santafly" seems to just have his version.
I suspect you’re right. Such songs wouldn’t be what everyone would consider “famous Christmas songs,” but they would certainly meet the OP’s criterion of “anything released by a recognized label.”
I’d seriously disappointed there are covers of Merry Christmas Darling by The Carpenters. You don’t mess with perfection people! 
I was facetiously going to suggest Tom Waits’ Christmas Card from a Hooker in Minneapolis, but to my surprise it turns out to have been covered a number of times by notable performers.
Mac Davis claims to have written something called “Drinking Christmas Dinner Alone” but I can’t find ANY version of it.
I haven’t found any covers of “Fairytale of New York” by the Pogues!
I don’t think Dan Fogelberg’s “Same Old Lang Syne” has been covered. It’s set at Christmas, though the title sometimes leads it to be a New Year’s song.