This is for any of the late night shows–Tonight Show, Late Show, Late Late Show, and so on. I think the weirdest bit I ever saw was on Letterman several years back. It must’ve been a very slow news day and I don’t think they even had any guests that night, so instead they played “Let’s Look For Rats.” They basically spent 20 minutes with a guy with a camera going around the Ed Sullivan theatre live looking for rats. I don’t think they have ever done this again.
There were a few visits from Brother Theodor to Letterman that were just surreal.
Of course, it could have been the drugs.
if strange=creepy, I would put Conan O’Brien’s “Preparation H Raymond” down. Actually a lot of Conan’s stuff might be considered Strange. Or at least complete non-sequiturs. One of the strangest things I saw was Farrah Fawcett’s interview w/Letterman a few years ago. Completely bizarre.
Did they find any rats?
One time in 2001 Conan did a segment at the start of May called “Sweeps Ahoy!” that was touted as a preview of the things viewers could expect from the show during May to draw in ratings.
One of these was called the “Late Night Cliffhanger.” If I remember correctly, a man in a bear suit was wheeled out on a fake horse, pointing a 1920s-style death ray at a dead fish that was lying on a plate. Menacing music played as the camera did close-ups on the bear, the horse and the fish.
They come back from commercial, and the segment basically repeats itself verbatim. Except at the end, a man dressed as a corn cob comes out and shoots himself with a hunting rifle.
My friends and I were doubled over in laughter. 
I don’t think so.
Another weird thing is what Letterman did just last week, when he had Biff blindfolded and running his hands over a wax Larry King.
Crispin Glover’s “kung fu” appearence on Late Night w/David Letterman.
Surreal doesn’t even begin to describe it.
From a recent Late Night:
Kermit the Frog, vomiting eggnog. What made it even more bizarre was that they didn’t bother to clean up after him, so the vomit was left to rest on stage, just in time for Kevin Spacey to throw one of the guest couches on top of it. Wow. 
They did! And it was a live feed, not pre-recorded (in-show, I mean). It was the high point of the evening. Hilarious.
When Jack Paar had the Tonight Show, he showed a mockumentary about spaghetti orchards. Italian-looking people were shown picking spaghetti from trees. Hundreds of fans wrote in to say, “I didn’t know that’s where it came from!”
The Conan O’Brian Show in claymation.
John Tesh playing “Turkey in the Straw” in a pen of turkeys, on stage, on Thanksgiving.
Max Weinberg having sex, missionary style, with Max Weinberg.
The Conan O’Brian Show in candlelight during the blackout.
I loves me some puffy-headed irish guy.
This was probably an American showing of an infamous BBC April Fool’s joke. The Brits were taken too, especially since spaghetti is a rather obscure dish over there compared to the US.
I couldn’t remember. How long ago was this, anyway? (I have a feeling I was 15 or so when I saw it, which would make it 7 or 8 years ago, but I could be wrong.)
Could you mean this: Top 100 April Fool's Day Hoaxes Of All Time ? First item, if I didn’t foul up the link (of course, Jack Paar could have borrowed it).
David Letterman has a recurring gag in which a Late Show employee “puts away the Late Show bear.” I just don’t get it.
Similarly, but going back to the late 1980s, earlier, funnier David Letterman show, Paul Schaffer did a musical, patriotic salute to…something or other. The salute was accompanied by red, white & blue tap dancers, and included a giant dancing hot dog. After the salute was done, Letterman asked Schaffer (paraphrasing)…
Letterman: “What’s the significance of the big hot dog, I don’t get that.”
Paul: “But Dave…I thought the hot dog was your idea!”
Letterman: “What? Wait a minute, who are you hot dog? What are you doing here?”
The hot dog, fidgeting nervously, runs off-stage and tries to escape from the studio. Security guards pursue him, and shoot the hot dog dead. It is last seen lying in a spreading pool of mustard.
Very, very, very bizarre. But at least it was funny.
I give it a max of 3 years. At the time, someone ran a story on the number of rats in NY City. It became news for quite a while (“how shockingly disgusting” type news) and Letterman decided to look for rats.
That was the time that Dave taped the show at 4am in the morning. Amy Sedaris was the only celebrity guest and she did a remote from her Manhattan neighborhood, where absolutely nothing was going on. The whole episode was a disaster, but a good effort, nonetheless. And as I recall, they didn’t find rat one.
Here’s one that sticks out in my mind: Letterman had a running gag where they would show footage of a barn owl eating a rat while playing “Tomorrow” from the “Annie” musical. The mismatch of these two elements was so hilarious to me.
Don’t forget the followup, “Max on Max, with Max watching.”
Actually, that was “Joel on Joel, with Joel watching.” And Joel Goddard has done a lot of really weird stuff on Conan. That weird smile gets me every time.
Don’t forget Conan’s beloved Masturbating Bear.
At about the same strangeness level as “Preparation H Raymond” was Letterman’s “Pea Boy” (from around the time of the NBC-CBS switch): he’d run out with a basket full of peas and throw them at the audience.
High on the Strange list are Chris Elliott’s appearances on Letterman during the 80’s: The Guy Under The Seats, The Fugitive Guy, etc.