This trope is being created per the A Worldwide PunomenonTRS thread. Please go there for the name discussion.
This work is drowning in puns. The characters are dropping them left and right, and often the narrator (if there is one) will get in on the action, too. Worlds of Pun generally capitalize on a quirky brand of humor, with the puns as an integral feature of its appeal.
See also World of Ham and World of Snark. Often includes Punny Names. See also Hurricane of Puns. For a character who frequently uses puns, see Pungeon Master.
Anime and Manga
Rumiko Takahashi's first breakaway success, Urusei Yatsura, is filled to the brim with puns -- its name, for example, can be read half a dozen ways depending on Kanji, Kana, and the use of spaces, each one of them a pun or joke.
In Sayonara, Zetsubou-Sensei basically everyone's name is a pun, sometimes complicated ones. This is ONLY pointed out in the case of the titular teacher and his family. Fansubs are kind enough to explain the name puns for characters introduced that episode.
One Piece lives and breathes puns. Attack names (most of Zoro's sword moves, notably, also resemble types of sushi when written), character names, and in the seventh movie over half the lines of the plot-central prophecy were puns.
Unglued and Unhinged are about 50% puns (the other half is a mixture of cardpaper and in-jokes that only players of the game will get).
Unhinged had Donkey Folk, which only existed to make puns on "ass". There was Smart Ass, Dumb Ass, Fat Ass, Cheap Ass and Bad Ass.
There are also the Clay Pigeon (a 1/1 flying bird that had an effect when thrown), the Rock Lobster (it wasn't a rock, and many take it for granite), the Paper Tiger (who burns bright and folds easily), and the Scissors Lizard (who has a lot of shear power).
Fowl Play turns things into chickens.
The Man of Measure is better at offense or defense depending on whether you're measured as taller or shorter than an opponent.
The Standing Army doesn't tap when it attacks, because they're always standing... but only as long as you are too.
The Spoils has got quite a few, too. Pick five cards at random, and there's a pretty high chance that at least one involves at least one pun.
Munchkin, in all of its incarnations. Meet monsters such as Dr. Jackal and Mr. Hide, Tequila Mockingbird and Cowthulhu... and that's only the beginning.
The Flemish series De Kiekeboes (a few English translations exist as Jo and co) has over 100 issues and in almost all of them at least one of the one-time characters, the companies or even the title itself is a pun.
A lot of Orient Men comics are based around puns, especially the last few ones.
The Astérix comic books are full of this, in both the original French and the English translations.
Airplane!: loads of them -- "Surely you can't be serious", Ted's drinking problem, smoking tickets.
Practically all the lines in Batman & Robin are "plant" or "ice" puns.
"Allow me to break the ice."
"I'm afraid my condition has left me cold to your pleas of mercy!"
Wild Wild West. Particularly the scenes where James West (black) and Arliss Loveless (has no legs) throw double entendre insults on each other's conditions.
Arliss: How nice of you to join us and add color to these monochromatic proceedings. --> James: Well when a man comes back from the dead I find that occasion to stand up and be counted. --> Arliss: Miss East tells me you're looking for General McGrath. I haven't seen him in a coon's age. --> James: Well I can see how it'd be hard for a man of your stature to keep track of... half the people he knows. --> Arliss: Perhaps the lovely Miss East can keep you from becoming a slave to your disappointment. --> James: You know beautiful women: support you one minute, cut the legs out from under you the next.
Scott: You once were a ve-gone, and now you will be gone.
Spaceballs. The radar gets jammed, they comb the desert, and of course there's the Druish princess.
Piers Anthony's Xanth books: There's been a steady increase in puns throughout the series. The first two books had only a handful of puns. After that, Piers Anthony started making the series more comedic, and adding more puns as part of the process. Then he started accepting reader-submitted puns and it and got completely out of control. Naturally, a great many Xanth fans were thrilled by this opportunity to actually be a part of their favorite series, even if only in a small way, so the puns flooded in ever-greater numbers, to the point that Anthony frequently has several books' worth of pun backlog.
Robert Asprin's Myth Adventures series has puns everywhere, including the titles of all the books and of the series itself. The various dimensions have Punny Names: Klahds are from Klah, Deveels are from Deva, Cupys (small, doll-like people) are from Cupid...
The book contains a sequence describing the ornamental armour Sam Vimes has to wear, and how it makes him feel like a class traitor. The pune-chline: "It was gilt by association."
And the Fat Mines contained BCBs (Burnt Crusty Bits) that Vimes said died because they were battered to death.
There's also an example of him being entirely unable to stop himself with the story of Fingers Mazda, who stole the secret of fire from the gods. He was unable to fence it, it was too hot. He really got burned on that deal.
Granny Weatherwax's lodgings in the Shades are made are all the better for being next door to a notorious reseller of stolen items. Because good fences make good neighbours.
Magrat believes that broomsticks are sexual metaphors when witches ride them. But this is a phallusy.
The name of the countries Djelibeybi and Hersheba. Terry Pratchett's realization that American audiences weren't getting the Djelibeybi pun inspired the creation of nearby Hersheba, which most audiences in general aren't getting. (If you've heard of the candy, the Djelibeybi pun is criminally easy to get, due to it being mentally pronounced the same way, andlampshaded when we're told Djelibeybi literally means "Child of the Djel." Hersheba is not as easy -- this is due to variation in pronunciation, the fact that it doesn't have a lampshade, and it doesn't have a book focused on it.)
The Phantom Tollbooth is one of these. There's a "watchdog" called Tock who is a dog with a clock in his abdomen. In the city of Dictionopolis people literally "eat their words" off plates. People literally jump to an island called Conclusions. It goes on and on like this.
Alice in Wonderland and its sequel, Through the Looking Glass, and What Alice Found There are both famous for their wordplay and allusions.
The band Alkaline Trio lives in one of these. Their albums have titles like Good Mourning and From Here To Infirmary. 
There's a band called the "Misfats" who cover Misfits songs and change the lyrics so they're about food. So they play songs like "Mommy, Can I Go Out And Grill Tonight".
Too many to go into, but during the early years of Dilbert, Scott Adams was REALLY REALLY into puns (very little office humor was involved, Dilbert was ostensibly an engineer just to provide a context for nerd-jokes and nerdy jokes).
Power Rangers. It's actually quite impressive to see how they can keep the pun ball in the air for long stretches.
Arch Knight Style Humor (adj) -- A savory blend of caffeinated epic failure served with with a side of cheese. This also probably means Artix wrote it himself... it is sort of like a train wreck. You really want to look away, but for some reason, you just... have to watch.
When Kingdom of Loathing isn't making random references to song lyrics, it's hitting you over the head with puns of varying quality.
"This is a bat with the body of a baseball. And the heart of a bad pun."
Pretty much every Sierra adventure game ever. The King's Quest series is probably better known for the puns in its death messages than anything.
Defense of the Ancients: All-Stars: technically found in Warcraft but in Defense of the Ancients the hero responses are heard constantly. Nearly every hero with a voice says some sort of pun. Nearly every response by the Pandaren Brewmaster is a pun.
There's a stage in Namco × Capcom, where every character is making lame puns. All of the puns are highlighted in red, to boot.
DevilBear uses puns frequently. The characters Bearalzebub and Lucy Fur in particular make puns the most in the series. Ursa, the Daiva of Wrath, however, indicates that she hates puns.
When Hejibits isn't doing a comic about Video Games, it's almost always a comic about a terrible pun of some sort.
Hybrid webcomic/browser game Demon Thesis does this constantly when in game mode, as virtually any action you have the characters take is accompanied by a pun or reference. For example, give Clady the spear and let her attack with it, and the attack is called "Clad the Impaler". Give Val, the sole American, the axe, and it's called "American Chopper" when she uses it. If Alain, a french-Canadian, goes into a defensive mode to take less damage, it's Dominic Deegan falls under this, especially during the early strips before Cerebus Syndrome set in. For example: Seer's catalog, bookmarking sites...
Almost every episode of House of Mouse used puns, especially in the cartoon shorts but often in the main story as well. Like Mickey saying he had to stop at the bank because he was "overdrawn" and two sketches saying "He's lucky! We're not done yet!"
Kim Possible is notorious for this, as Kim herself has noted once or twice.
Five hats means that five tropers think it is ready to publish.
You are saying that you think this draft is ready to be published. That means the description is not ambiguous,
it doesn't duplicate an existing trope, there are at least three examples, and the title makes sense.
Is that what you meant to do?
You are saying this draft has a ready-to-publish hat it does not deserve and you are taking it back.