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A character that is all-powerful. Either literally omnipotent, or are simply so powerful that they're virtually omnipotent within the context of the story.
There are several types of omnipotent characters.
The character has absolute omnipotence. As in, nothing can challenge them, and they can literally do anything and everything, logic and causality be damned.
The character is omnipotent. They can break even logic and causality to achieve anything, but there are others who can do this as well (sometimes they're The Anti-God). What happens if they go against each other can be a massive Mind Screw.
The character can do just about anything that's logically possible (i.e. virtual omnipotence), or at least appear to, thus making sure Magic A Is Magic A in the process. This type of omnipotence is also more frequent than the above type to have multiple characters that operate at this level. The term nigh-omnipotent is often thrown about at this level.
The character is omnipotent within a field or concept. Specialized omnipotence, if you will. Anthropomorphic Personifications are almost guaranteed to display this with the concept they embody.
The character is "merely" an almighty being, whose power is far beyond any other in the setting. For practical purposes though, they're basically omnipotent.
In A Certain Magical Index, Othinus is a Physical God, but has a weakness that her powers only work 50% of the time. Once she overcomes this, she becomes virtually omnipotent, able to rewrite reality of the entire universe on a whim. Her only limitations are that she cannot affect Touma Kamijou (due to his Imagine Breaker) and beings who exist in a state between life and death directly. She also cannot reverse her changes to the universe; she can only change it again ad infinitum, though she was able to change the universe back to normal with the help of Imagine Breaker. Contrary to her claims that she's far above humans, she still has a human-like mind: she almost collapses from mental fatigue at one point.
Haruhi Suzumiya, unbeknownst to herself. And considering her behaviour, it's probably better for the SOS Brigade to keep all paranormal stuff away from her in secret, lest..
In Tenchi Muyo! we have The three Choushin: Tsunami, Tokimi, Washuu, and Misaki as type 2, and Tenchi himself as type 1, though not manifested as such yet. Z and Ryoko at full power are arguably type 3. The Zinv and current Tenchi are type 5, as are the most powerful Jurai trees.
In Medaka Box Ajimu Najimi is functionally omnipotent, in that she can do anything including changing the laws of reality, but feels bound by the Theory of Narrative Causality which prevents her from ever winning against a "main character".
Hao Asakura has become the lowest example of this in Shaman King before the heroes even meet him. His control over reincarnation, connection to the Spirit of Fire, and centuries of experience mean he's unbeatable from the get-go, with the next-best shamans being as nothing in comparison to his power. In a setting where resurrecting a dead person is possible, if draining, for the very powerful, Hao can perform mass resurrections after single-handedly defeating a human navy supplemented by powerful shamans. The plan to defeat Hao is to let him win the tournament and gang up on him with the five Olympus Mons elemental spirits while he can't fight back, but the heroes admit even this is fairly unlikely to work. When Hao awakens as Shaman King, he's omnipotent and kills the entire cast with a thought as he prepares to reshape the world to better please himself; fortunately, Yoh expected something like this would happen when he realized even if they were strong enough to win, the heroes would never be able to end the threat Hao presents with violence. Instead, the entire cast appears within Hao's mind and manage to avert the extinction of ordinary humans by reasoning with Hao instead of trying to overpower him.
Puella Magi Madoka Magica: The titular character was likely somewhere on the spectrum as a magical girl in the penultimate timeline, at least until she paradox'ed herself out of existence. She was powerful enough to restart the universe with new laws of nature in place to make it a little less Crapsack, and annihilate her own Witch with a single, devastating blast.
Tengen Toppa Gurren Lagann: The Anti-Spiral is not quite absolutely omnipotent, but it comes close. It can build universes, manipulate probability, and matches everything the heroes dish out with little visible effort (except for Simon's last attack, which kills it). Entire galaxies are mere tools in its hands (literally).
The titular character from Enigma is what happens if God was gay, morally ambiguous, and ate lizards. It's kind of a weird comic that way.
The Beyonder from Marvel Superheroes Secret Wars and Secret Wars IIbelieves this because as he living embodiment of a universe, he can do many things. He was later retconned into something less impressive, but until then he was nearly omnipotent.
In Marvel, a supposed characteristic of the One Above All, who is also hinted to be the capital-G God. The most powerful entity stated that appears, the Living Tribunal, seems like this compared to the protagonists but it can't act without all three of its heads in agreement. So it usually does nothing.
The DC Presence is, or was, occasionally referred to as The One Above All by some, including The Spectre. This was a probably deliberate hint that the Marvel and DC God are actually the same being. In other words, He is the most powerful comic book character period. Assuming He isn't simply code for "the writers".
Also, The Presence is the same being as The Source from the New Gods (probably), as both are aspects of the same all-powerful being. Other aspects exist (eg. The Voice), but they are mentioned far less often.
Man of Miracles or M.O.M. for short in the Image Universe, or at least in Spawn, s/he is also Kali and Jesus Christ, and God of the Old Testament is just his/hers son.
Mr. Mxyzptlk is a Superman villain from the 5th dimension who's abilities makes him a Reality Warper in the 3rd dimension. Fortunately, he's more interested in messing with Superman and playing tricks than doing anything really malicious.
A pair of supposed omnipotent entities in the Marvel Universe once got into a debate over this when they ran into the the Celestials. One of them didn't understand why the other was so intimidated by the Celestials because they were both omnipotent and nothing should be able to threaten them; the other told her that yes, they were omnipotent, but there are "levels of omnipotence" and the Celestials are on a much higher level than either of them.
God in Bruce Almighty seems to be this. He then makes Bruce omnipotent too, aside from not being able to affect free will. There doesn't seem to be anything he can't do. God left out giving Bruce omniscience in the package however, so he's still limited by his human mind and eventually humbled by how he failed to do a better job carrying out God's responsibilities.
The Ellimist and Crayak from Animorphs both have unlimited power, and regularly cause the creation or destruction of species. While they agreed to no direct interference as part of their "game," neither of them fit the All-Powerful Bystander trope - they both are skilled enough at manipulation to still cause massive changes in the history of the galaxy.
Happens at the very end of Brain Jack by Brian Falkner. Sam ends up merging with the A.I. Ursula and gaining the ability to control everything and know almost everything as well, because in this future, Everything Is Online.
In Heir to the Stars: Overture by Lionel Suggs, there are more than a few of these types of beings in the story. There are several of characters that are the Be-All, End-All of levels of existence. In the very opening, it states They are defined as the Be - All, End - All of the perpetual, unchanging, unbounded, immanent, and transcendent truth, which are the existences and principles behind the Divine Ground of all matter, energy, time, change, action, necessity, fate, destiny, reality, space, being, actuality, potentiality, possibility, nothing, the active principle behind the phenomena of the absolute infinite and imagination, and everything above, beyond, and outside a system of everything beyond The Mainfold. It even goes in to explain that there beings known as Overtures and Finales. An Overture and a Finale can do anything they supremely well please because they are the Grand Principle behind the infinitely All-powerful, THE ALL, any and all systems, as well as any and all states; encompassing all meanings of Omnipresence, Omniscience, and Omnipotence..
The Incarnations of Immortality have the "specialized omnipotence" variety: When they assert themselves in their particular domain, no one has the power to oppose them, not even another Incarnation.
Nobody in Starfleet possesses Complete Immortality, apparent omniscience, or the ability to warp reality with a click of their fingers, so that's being a little overly modest. Unlike Starfleet, whose powers are dependent almost solely on technology, the Q's powers are inherent to themselves. The only known limit to their powers is the rest of the Continuum, who can collectively impose punishments like removing their powers (though they still know almost everything), imprison them or make them mortal. That and a Patrick Stewart Speech.
Also the Douwd, a being who casually exterminated an entire race in an instant in a moment of anger. Probably a type 2 or 3. While he can transmute matter and simulate life, he does not appear to be able to control space or time. Thus he could neither simply send the Husnock attackers someplace else, nor truly resurrect his wife, the other colonists or the Husnock after they were dead.
The EU novels have 0, a former friend of Q who is arguably more powerful than Q, but in the past was crippled to not be able to travel in time or faster than the speed of light.
Nobilis: type 4. Nobles can do virtually anything within the confines of their Estate, especially if they invested heavily in Domain and Persona, which govern (respectively) your control over the raw material of your Estate, and your control over the little "[Estate]ness" dial on everything else in the world.
The position that God can do anything that is logically possible (while still possessing absolute supernatural power), but cannot do things that are mutually exclusive, like making a stone so big even He can't lift it, is generally the position taken by some Christian theologians and philosophers such as St Augustine or St Thomas Aquinas- God can do anything, except that which makes God "not God", so He cannot do something that removes or defies His omnipotence, including simply making himself not omnipotent (since one paradox is that an omnipotent being can make himself powerless, but still be omnipotent, since he is omnipotent- these philosophers say that He can't, thus removing the paradox). For Augustine, this is because God is perfectly rational, and is also perfect, and being perfect would not act contrary to His nature, which is to be rational, and since such paradoxes are irrational He can't do them. In other words, the question is regarded as fallacious and nonsensical, and just a trick of word play.
On the flip side, other Christian philosophers like Rene Descartes say that God can do absolutely anything, reasoning that He existed a priori to logic and is the one who tells logic what it is. The only reason such paradoxes exist is because logic acts a certain way from a human perspective, but the level of God is one humans cannot comprehend and what are apparent logical paradoxes to us are not so to God. This is called Absolute Omnipotence. By this thinking, God could (to use the above example) create a stone so large not even He could lift it. And then lift it anyway.
Genies. Most mythologies/folk tales about them indicate the only restriction is that they are bound to their lamp/jar/whatever, and must obey the one who frees them (and this gets subverted sometimes). Some works add rules for plot convenience like Aladdin, but generally a genie = Phenomenal cosmic power, itty bitty living space.
In folk tales and other fiction, they seem to be omnipotent. In mythology, however, they're supposed to be able to do many things humans can't, but also unable to do many things humans can, and often aren't bound to lamps or anything else.
Averted by most Polytheistic pantheons, since even the Gods are subject to fate.
In BioShock Infinite, by the end of the game, Elizabeth becomes essentially the Omnipotent of that universe; she is able to "see behind all the doors", meaning she has the ability to know anything in any dimension or timeline. She is also able to open portals to any dimension or timeline, and it's implied that she becomes immortal and frozen in time, like the Lutece twins (this is NOT confirmed, though, it's only a speculation.)
Asura's Wrath plays around with this: Chakravartin is referred to as the Omnipotent Ruler of Gaea. However, he's eventually defeated by Asura.
Umineko no Naku Koro ni has Featherine AugustusAurora, who is perfectly capable of pausing the plot of the visual novel she's in, deciding she wants a scene to play out a different way, and rewriting the script of the novel itself. The one 'fight' we see her take part in, she decides she can't be bothered to think of a fight scene at that moment, so writes down that Lambdadelta gets defeated by [Blank Space], saying she'll fill it in later. This results in Lambdadelta being killed by nothing, out of nowhere, instantly. The character who got one-shotted was laughingly throwing and tanking big bangs, and had, up until that point, been the tied-for-most-powerful character in the whole story.
Turning the player character into this is the reason it's called "God Mode".
Scribblenauts protagonist Maxwell has a notebook that grants him the power to create practically everything conceivable to the human mind aside from things that are inappropriate (which as of Unlimited can be circumvented through object creation), including adding the adjective "Unbeatable" to himself which prevents him from receiving harm, even by Death's instant kill ability.
minus. is about a seemingly omnipotent child, who mostly uses her power as a toy.
All of the Author characters in Bob and George have god like powers. Seeing as how The Author is the creator of the comic universe he IS a god there. The Helmeted Author is apparently an Author of a different universe and the Shadowy Author claims to be one at first and ultimately turns out to be a future version of the actual Author.
In Homestuck, every First Guardian is omnipotent, and can also be omniscient as well depending on what they were made with. Other characters are not First Guardians but gain their powers for themselves. Abilities shown so far:
Schlock Mercenary: A certain AI named Petey (who was already insanely smart, and already controlled a huge fleet of warships) manages to take over a power generator made out of the Galactic Core! Except for the Pa'anuri, there is practically no one who can fight him... or even last three minutes against his moderate dislike.
One of the terms coined by the Spacebattles forums is "ROB," or Random Omnipotent Being. A ROB is used to justify ridiculous scenarios, or create events for people to talk about. A ROB is usually used as a Deus Ex Machina for a random event to set the plot moving in a quest.
In Funny Business, Jeannette has no limits on her power. Lewis takes advantage of this to solve the omnipotence paradox.
In one of the shorts of 'The Simpsons'', Treehouse of Horror (the theme being dreams resulting from overconsumption of candy), Bart is an omnipotent boy who freely changes things about him for his own amusement, the whole episode being an homage to a Twilight Zone episode. After turning Homer into a jack-in-the-box, the family take Bart to a psychologist and helps Bart and Homer bond, resulting in Bart waking up from the dream.
Amazo after his/its return in Justice League Unlimited has learned or copied the superpowers of every entity in the universe. The only power he lacks is imagination to find a purpose for all his powers. That's why he is coming after Lex Luthor, his surrogate father-figure, next. However, Chaos Magic is something he cannot deal with, as it can absorb his attacks — he left the Earth in fear when he found this out, to figure out a way to counter it... and as far as we know, he is still doing just that dozens of episodes later by the time of the Grand Finale. A little embarassing really, since the League solved the problem themselves a few minutes after he left.
Discord of My Little Pony: Friendship Is Magic can use magic to do pretty much anything at will, up to affecting celestial bodies — fitting, as he's voiced by John de Lancie, who plays Q in Star Trek (as noted above). He even snaps his fingers to work his magic, and with the same sound effect when things do happen. The only thing more powerful than him seems to be the the Elementsof Harmony. He's not always immune to or able to overcome magic used by others in specific cases, such as when Princess Celestia casts a spell on the Elements to keep him from just magicking them away, but this is rare and always specific. In addition, when Tirek absorbed the magical energy of hundreds of pony, he grew powerful enough to where he could drain Discord's magic.