“Nobody but me” is mighty strong. But if I had to pick my favorite (relatively) obscure song by a variety of artists (or, at least, songs I never hear on the radio and that the acts never seem to perform live), I guess I’d name:
Aerosmith: Lightning Strikes
Cream: Dance the Night Away
Electric Light Orchestra: Summer and Lightning
Kinks: Shangri-la
Led Zeppelin: I’m Gonna Crawl
Pink Floyd: Fearless
Scorpions: I’m Going Mad
Van Halen: On Fire
Yes: Future Times
ZZTop: I Got the Message
I agree they’re not hugely obscure, as I took the title as a bit of hyperbole: Trying to imagine something that only you have heard of, which isn’t something like someone singing in the shower or some similar very private event, is futile, and the Internet is good at proving it futile. 
So I pitched my post at less “back of the Goodwill shop ultra-oddities which showed up on WFMU once in 2002” and more at stuff that almost hit the mainstream but has since been mostly forgotten except among the true fans.
No, but I should.
Ha! I’m not sharing my top secret song with the likes of you!
Oh, okay, but don’t tell anyone else.
Van Morrison - All Work and No Play
The youtube with the highest # of view for this song is ~8000, whereas “Moondance” has over 6 million.
Hey, lay off WMFU! Plus, my band’s record has been played on there around a dozen times, not just once, thankyouverymuch! 
I love WFMU! I once had a college radio show called the Stochastic Hit Parade, and I’ll forever remember how that station introduced me to streaming audio and all manner of weird and wonderful music, such as “Zombie Jamboree” by Lord Jellicoe & His Calypso Monarchs, “Forgive Me, John” by Jean Shepard & Ferlin Husky, and “I Wanna Be A Country Singer”, by Bill Carlisle And The Carlisles.
Lumpy Beanpole and Dirt…bad livers…aka Danny Barnes and Mark Rubin…
ENJOY… (i shout sometimes)…
again ENJOY!!!
tsfr
That may be the funniest novelty record ever.
and on preview: Hey! I’ve met Mark Rubin! We have mutual friends, he’s great. My favorite from the Bad Livers is Hogs on the Highway
Hall and Oates’ Italian Girls on H20.
Turn This Thing Up from Brassy.
Diving into the WFMU archives again reminded me of two more tracks:
Linda Laurie - “Ambrose (Part 5)”: In keeping with my previous picks, this is a novelty track from 1958 which never got played much on nostalgia stations. Maybe because she did both voices, and at seventeen, no less.
Tom Waits - “What’s He Building”: There are whole genres of TV show and video game which would sacrifice interns to become half as atmospheric as this track.
My personal faves:
James Taylor on Howard Stern playing Woodstock Acoustic
As long as we’re on an acoustic kick…James Hetfield playing Mama Said solo Acoustic
Speaking of James Hetfield and playing acoustic…Beatles In My Life
and just saw this for the first time…Metallica covering Brothers In Arms Acoustic