Monty: Do you always carry a glass cutter around with ya?
Gadget: No, only when I wanna cut glass.
Considering that Dora the Explorer is formatted in a style reminiscent to a point-and-click adventure computer game, the fact that Backpack is able to carry more than it can logically hold is a possible Homage to this trope. Its Spin-Off, Go, Diego, Go!, also does this with its own inventory character, Rescue Pack.
In episode "Fear of a Krabby Patty," a frantic, delusional SpongeBob gives Plankton just enough time to squeak out, "Hey where'd you get that piano?" Just before, well, you know.
Plankton did one of these himself in his debut, "Plankton!", offering SpongeBob a golden spatula in an attempt to get a Krabby Patty from him. "I've been keeping it in my...secret compartment. SHING! Sparkle-sparkle."
In "FUN" where he learned FUN from SpongeBob. He managed to hide the Krabby Patty in his pocket. How does he even have a pocket, let alone hide something more than two times his size?
In "Christmas Who?" Spongebob grabs pencils, paper and a photo frame from nowhere.
No matter how many times Tucker's PDA gets destroyed during the plot, he always has another one ready next time. In the 3rd episode, Sam asks him, "How many of those things do you have?"
Then there is the Fenton Thermos. Sure, sometimes Sam or Tucker will arrive on the scene just as Danny decides he needs one, and the animators may feel adventurous enough to show him carrying it on his back, but on most occasions this thing comes out of nowhere. It is unclear how many Fenton Thermoses the Fenton family actually owns. Considering the Fentons have a weapons vault, I think it's fair to assume that they have mass produced the Fenton Thermos. They are Crazy-Prepared for ghosts, after all.
A strange variation of this happens with Danny's alter ego. In one episode, Danny walks out his front door with his backpack on and transforms into his ghost half to fly to school. The backpack just disappears (until he turns back into his human half, that is)(this can also be applied to his outfit when he switches from one form to another).
In the same episode, he pulls a piece of toast out of nowhere in ghost mode.
A headcanon this troper supports is that Danny has a pocket dimension he can access to store small objects (maybe not consciously) and that is where that stuff goes. After all, he got his powers from an interdimensional portal!
Also a Running Gag from the original comic and throughout the adventure games.
Wreck-Gar in Transformers Animated has his trash bin, that functions as both Hammerspace and a "Pit of No Return". He can take many useful (and more useless) items from it, and things that are put in it seem to just disappear. Swindle's chest compartment does the same thing, except it canonically links to a hammerspace.
In Transformers parlance, hammerspace is known as subspace and is used for a number of things, including weapons storage and a place to shunt mass when transforming into something tiny (for instance, when Megatron becomes a gun). And yes, this is also where Prime's trailer goes. And long before being largely canonised as subspace, this mysterious dimension was actually known to many fans as "Trailerland".
Some continuities from different writers (mostly the comics) have been attempted to justify it with actual talk of mass shifting technology, but in the end it is just A Wizard Did It. The Transformers films did everything it could to avoid this, with appropriate vehicle modes to accomodate robot modes (with debate about Frenzy's head turning into a cell phone). In fact the All Spark as being capable of this was presented as a very unique and special thing it can do.
The Minicons (at least in the Cybertron series) seem to have the inverse of their larger counterparts. In their vehicle modes they are nearly the same size as any other Transformer, granted a tad smaller; Jolt in particular stands at least 10 feet tall in his helicopter mode and has enough room inside for two full-size adults to ride along. In his robot form, Jolt rivals the kids in height and can not only enter houses, but one episode had him hiding in a closet with another Minicon of the same size! Their cockpits seem to function like this as well. When Vector Prime finds Atlantis the kids use the Minicons to travel underwater; the Minicons transform into their robot modes underwater while the kids are still inside them, apparently with plenty of room for them to gesture wildly when startled. It isn't until later when VP drains the water out of the city that the kids come out by just walking out from behind the robots which are at head level.
In the cartoon series Popples, hammerspace is at their back and call when they reach into the pouches on their backs, especially when the Rule of Funny is applied.
In Count Duckula, Nanny was able to produce any item from inside her sling.
Duckman: When Cornfed reminds Duckman that everyone in town is working only a half-day (due to a civil defence drill that our antihero didn't hear about), Duckman says, "Ohhh, it's one of those days," and throws on — pulled out of nowhere and in a fraction of a second — an ultra-Orthodox Hasidic Jew's black hat, coat, and sidelocks. (Cornfed's response, just for the record: "Duckman, this isn't one of those days where you have to pretend you're Jewish.")
Abu the monkey from Aladdin: The Series has hammerspace inside his vest; he can hold any amount of stolen money, jewelry, fruit and the genie's lamp inside it.
During the second season of X-Men, after having lost his ultimate telepathic powers, Professor Xavier is able to produce weapons instantaneously in order to save Magneto, including a boomerang, and a spear!
Done countless times in Ed, Edd n Eddy, with examples being Kevin pulling out a bike from his pants in "See No Ed", Rolf pulling out a giant hammer out to hammer Double-D into the ground in "Momma's Little Ed" and in "A Fistfull of Ed" with Eddy pulling out hotdogs and eating them (the kids even start asking themselves where Eddy got all those hotdogs in the first place).
Marge's hair appears to contain hammerspace, as she has retrieved various items from it. On one occasion whilst swaying from side to side doing exercises, each change in direction dropped a larger item out of her hair.
Krusty's small car can also transport several clowns, each wearing oversize footwear.
Phineas and Ferb, "The Chronicles of Meap", lampshades it when Candace calls out adorable alien Meap on "where he keeps all those pictures" he uses to communicate.
Lampshaded in an episode of Timon & Pumbaa, where the title characters fall off a cliff and pull increasingly ludicrously large items— pianos, elephants, cruise ships— from behind their backs ("You never know what might happen to be just behind our backs!") while Rummage Failing for a parachute.
The characters in Shirt Tales were capable of pulling random items out from under their shirts until something useful popped up.
Courage the Cowardly Dog revels in this. At least five times in a single episode, characters will pull oversized items (giant mallets, giant masks, even loose change the size of frisbees) out of their pockets or from behind their backs. In fact, every cartoon Jon Dilworth has made does this.
The characters in the Men In BlackAnimated Adaptation series sometimes pulled some ridiculously large weapons from under their suit-coats. In one episode an MIB not only drew a large weapon for himself, but then pulled out another to toss to an unarmed comrade.
In Kiwi, the bird pulls out an actual hammer, plus nails, from seemingly nowhere.
Lampshaded in Drawn Together when Wooldoor says he keeps things in his fanny so he can "pull them out for comedic effect".
Agent Six from Generator Rex keeps fold-out katanas in his sleeves. He also periodically produces throwing knives seemingly from thin air, but at least those are pocket-sized.
Gorga, the pet alien dog from Zula Patrol, has a Hammerspace up its nose! Or is that its mouth?
On The Hub'sPound Puppies, the dogs present new adoptees with collars as a parting gift, and quite often, they do so away from the pound itself. The team could be pulling them out of their fur, however, all of them have presented collars. Only two of the five members are shaggy enough to possibly hide a collar in their fur out of view, with only one of the five not having a visible collar of his own.
Like Inspector Gadget, Hanna-Barbera's Captain Caveman has a Bottomless Magazine of objects and creatures he can pull from his body-length body hair, from a skeleton key to a large, bone barbell (which he presses with ease). He even pulls things that are far larger than he is out of it sometimes, like a large dinosaur that he and his friends use like a stairway.
Taz-Mania used many of the tropes from the Looney Tunes shorts, so Hammerspace was well and truly in effect. In "A Flea for Me", for example, the flea pulls a complete oil well out from under his coat.
The eponymous character of Mr. Bogus seems to keep an unlimited supply of assorted items in the pockets of his pants, given the occasion that he would reach into his pockets for an item that befits his current situation.
In the Archer episode "Skorpio", Archer, naked after a rather messy three-way sexual encounter, literally pulls a grenade out of nowhere to throw at some mooks. This is even (literally) lampshaded immediately afterward, when Lana asks him where he got the grenade, and he responds by saying that "It was hanging from the lampshade!"
Aloysius Pig of Garfield and Friends' B-segmentU.S. Acres can pull items from the back of his jacket, despite not having a pocket on the back. He only did it twice: once for a picture of smiley faces (which he called a whole staff of artists) and his business card.
Heckle and Jeckle are locked in prison in the cartoon "Super Salesmen." Prison guard Dimwit discovers Heckle using a metal file to try to escape, so he confiscates it. Heckle takes another file out of his prison cap, which Dimwit grabs, followed by a succession of files. Dimwit grabs the hat, turns it over, and whole pile of files pours out.
My Little Pony: Friendship Is Magic frequently features the use of hammerspace by ponies, but mostly in small ways. Pinkie Pie, however, very frequently produces things out of Hammerspace, though Rarity has a slight tendency towards it, especially with her drama couch (for her to fall on; pulled on from just offscreen) in Lesson Zero. Spike occasionally digs around in his pockets for some items, even though he doesn't wear clothes.
In Equestria Games, after melting the ice cloud, Spike pulls an umbrella out of nowhere.