Ryuk from Death Note is most definitely a Great Gazoo. His only real manipulation is letting humans think they are in control of the situation as part of his Screwy Squirrel routine. He is very potent, impossible to harm conventionally, and the only way to defeat him would be manipulate him into sacrificing himself for someone else's sake - something extraordinarily unlikely given his carefree attitutde.
Excalibur from Soul Eater, the most powerful weapon on earth and also the most obnoxious and pettily selfish.
Dung Beetle/Koyemshi from Bokurano counts as well. Proves himself generally invulnerable to the actions of the main cast.
Mr. Mxyzptlk, an imp from the fifth dimension. In Superman: The Animated Series, Mxy claims to be the inspiration for the stories of genies and leprechauns - after a while he got bored of messing with ordinary people, but then Superman showed up. It's unclear just how far up the ladder he is on his home turf (sometimes he's unknowable, sometimes he's a 5th-dimensional kook with a weird hobby), but in the third dimension he's a Reality Warper. Supes can only get rid of him by tricking him into saying his name backwards - a weakness that, in Post-Crisis continuity, Mxy made up himself because a game has to have rules; Pre Crisis, it was a naturally ingrained weakness. In The World's Greatest Superfriends, he even tricked Wonder Woman, Aquaman and Supes into making the ingredients needed for a potion that'd enable him to overcome his weakness but Supes wised up and not only foiled the plan but also allowed Mxy to think the plan worked to trick him into saying his name backwards. When he's sent back, a Reset Button is hit, and everything returns to normal.
In the Golden Age, Mxyztplk (note the spelling) was in his own dimension a mere court jester to his world's king. Also during this era, saying his name backwards wasn't just his own weakness; anyone who said Mxyztplk's name backwards would be sent to the fifth dimension.
One Silver Age story had Superman turn the tables by traveling to the Fifth Dimension and using his superpowers to prank Mxy exactly the same way he normally does on Earth. Mxy attempts to get rid of Supes by getting him to say "Namrepus", but it doesn't work; after he decides he's had enough fun, he goes home by saying "Le-Lak".
Bat-Mite is to Batman as Mxy is to Superman...except he's a gigantic fanboy who honestly wants to help instead of cause trouble. Naturally, Batman finds this even more annoying than if Bat-Mite were just out to get him. (Also, Bat-Mite does sometimes knowingly make things harder for Batman, just to see how his hero is going to get out of the situation.)
Johnny Thunder's thunderbolt (actually a being from the same dimension as Mxyzptlk).
A minor recurring nemesis of the Fantastic Four was a shape-shifting alien troublemaker known as Impossible Man.
In the past DC had Mxyzptlk imply that he was visiting the Marvel universe in the form of Impossible Man, but the Marvel Handbook says that this isn't the case and Mxy is merely imitating Impossible Man. This was further disproven in the Superman and Silver Surfer crossover. Mxy and the Impossible Man teamed up, and later started fighting. And Mxy was quite offended at the thought of being equated with the Impossible Man.
It also highlighted some of the chief differences between the two: Impossible Man just loves to have fun, while Mxyptlk's humor has a darker edge to it. The former actually gets enraged when he realizes Mxy lied to him.
With the popularity of Mxy and Bat-mite, it was planned to give The Flash a helpful imp by the name of Mopee. However the Retcon involved was so hated that the very next issue ignored the entire thing. He was effectively out of continuity for decades, though recently appeared in two issues of the DC Super Friends series. During which he claimed to not only have given powers to the Flash, but also Superman, Wonder Woman, and Batman. No one buys any of it. He also gets his first taste of the troublesome side of the trope, giving everyone at a convention super powers to prove he really can do it. Naturally Myx planned the whole thing.
Even Aquaman has an imp related to Mxy, the Thunderbolt and all the rest, by the name of Quisp. He is best known for turning up during the Grant MorrisonJustice League of America run, having made himself absurdly Darker and Edgier to better mirror Aquaman at that time. It's fairly clear that Morrison intended this as a parody.
Issue #65 of The Powerpuff Girls introduced the Micro-Puffs, three sprite versions of the girls from another dimension. They first appear to want to be friends with the girls but their ulterior motive is to yank their collective chains with mischief.
A far more malevolent version is the X-Men's enemy Mojo. He's from another dimension that he rules absolutely through a brain-numbing media empire, although he can only maintain ownership of the Mojoverse so long as his subjects like his programming. To that end, he uses his utter and complete mastery of magic (or sufficiently advanced science, maybe) to irritate the X-Men, so they have wacky adventures. He rarely attacks them outright; he does, however, have an army of baby clones of both the X-Men and of their enemies, including those of the Age of Apocalypse. Ominously, Dr. Strange once remarked that if Mojo ever decided to appear on earth itself, it would be a very, very bad thing...
Films — Animation
Disney's Aladdin has the zany Genie. Virtually any good magical character in Disney is typically pretty wacky; the Three Good Fairies, the Fairy Godmother, Peter Pan, etc.
The Djinn from the Wishmaster films is a far more malevolent version.
Lone Wolf: Really the only way to describe Alyss, a mischevious demigoddess first introduced in the novelization before becoming a Canon Immigrant in the gamebooks. She's firmly on the side of good, but is rather playful about it compared to any other of Lone Wolf's allies.
Tom Bombadil, in The Lord of the Rings. Tom is so powerful that he is able to wear the One Ring and give it up freely without a thought, but spends most of his time wandering in the Old Forest and whimsically singing about himself, and is said by Gandalf to be capable of withstanding an assault by any force of the world, unless they were all allied against him. He provides a convenient plot device to allow the hobbits to escape the Barrow Wights and take their swords (which later prove essential to the story, since the swords are the only thing deadly to Nazgûl).
Let's just say that Bombadil really doesn't fit neatly into Tolkien's larger backstory and there's a lot of fannish speculation about who or what he really is. Gandalf implies that Bombadil being unaffected by the Ring is just that Tom doesn't care about the Ring (beyond a brief passing interest in it as a pretty shiny thing), so it has no power over him one way or the other.
Simkin from The Darksword Trilogy is a rare example of a human-looking Great Gazoo. He spends most of the books simply inventing his own plotlines if he doesn't like the situation he's in.
Pennywise the Clown from IT is a malevolent form of this trope. Wacky, powerful, and completey evil and murderous, and invisible to most people.
Paladine in the Dragonlance novels is a good version of this trope. Consider how every single time Tasslehoff prays to him in the Legends trilogy, Paladine immediately gives Tass exactly what Tass asked for, never quite what Tass had in mind, but exactly what Tass actually needed.
At Epcot in Walt Disney World, the Imagination Pavilion has the excitable purple dragon Figment. In the original incarnation of the Journey Into Imagination ride, he used his reality warping powers to come up with new ideas. In the current incarnation, he mainly uses them to annoy Dr. Nigel Channing (played by Eric Idle) in an effort to get him to lighten up and let his imagination run free.
Thanks to the great powers and incomprehensible natures, certain Transapients and Archailects have been considered to be these in Orion's Arm, in the eyes of ordinary modosophonts.
The Gargoyles incarnation of Shakespeare's Puck. In his first appearance (but not later ones,) he's bound by an enchantment that makes him a Jerkass Genie.
The Great Gazoo is given a Shout-Out in The Simpsons' "Spin-Off Showcase" episode, showing "Ozmodiar" as a little green alien that only Homer can see. The joke many fans missed was that in an interview, Matt Groening said that the introduction of a Great Gazoo would officially mark the death of the series.
In one episode, the Simpsons are at an animation convention, and Bart and Lisa watch a clip from an Anime:
Orko from He-Man and the Masters of the Universe (1983). The twist? He only has omnipotence in his homeworld — in Eternia, he is mostly an Inept Mage. However one episode features another trollan called Prankster, who is an even straighter example, and he makes a lot of trouble for He-Man and co. The only way to get rid of him was to trick him into saying his real name.
The 2002 reboot at least demonstrated why this was. Orko could manage his powers just fine with his wand, which he accidentally lost while saving a young Prince Adam and Cringer shortly after arriving in Eternia.
There's an episode with a blue cat creature named Chaos who was said to be stronger than any genie.
Another episode had a pair of small gazoo-like Reality Warpers, who force the cast to compete in various games. At one point Iago suggests that, with their powers, they should try to Take Over the World or something instead. They reply that ruling it got boring after a few centuries.
Princess Celestia is a Downplayed Trickster Goddess. Despite being capable of dealing with most threats to Equestria herself and/or providing far more information about said threats than she does, Celestia usually sends a (seemingly) ill-equipped Mane Six instead despite the (seemingly) impossible odds. This is to train Twilight.
The first time involved a Take That to Wesley Crusher on Star Trek: The Next Generation with the network executives adding in the annoying, space banjo playing Snirkles to make Wesley look better in comparison. The fans paid for a billboard to read "Kill Wesley. Keep Snirkles".
The second time actually had Gazoo appearing to the Flintstones for the first time. Fred and Barney believed him to be some type of god and offered their wives and animal sacrifices to appease him. They then savagely killed him when they get sick of him.