Virlomi, of Orson Scott Card's Ender’s Game saga, believed that she could communicate with the gods, and that she was divinely ordained to save India. All of India worshipped her as the Goddess of the Bridge.
Orson Scott Card's Homecoming series: The Keeper of Earth is never outright stated to be God (or even Gaia), but with the stunts She (as The Keeper is usually called) pulls off (sending an image of Nafai's face to a Digger girl 1,000 years before he was born, chasing the bulk of humanity off with a spontaneous Ice Age), She might as well be.
If you want a webcomic approximation of this, check out Bob and George.
Frank Herbert's Dune, the protagonist, Paul Atreides becomes the Kwisatz Haderach, the universe's super being. His consciousness can be in many places at once and can see things before they happen. He is worshipped as a messiah and god.
An important point that The Film of the Book apparently missed: Paul is not a god (the limitations of his power are repeatedly shown), and doesn't consider himself so. He is a man playing on superstitions to appear as a god to his followers. While he denies his own divinity he states plainly that his sister (Alia) is a goddess. Of course this may have been for effect.
Now, his son, on the other hand... well, there's a reason the fourth book is called God-Emperor of Dune. Paul could have become an actual godlike being (by merging with sandtrout to become a human-sandworm hybrid)), but he could not bring himself to so fully sacrifice his humanity. But Leto II (who shares his father's abilities) sees that for mankind's future, he must.
Unlikely that Paul could have achieved the transformation. Remember, Leto was born with the spice in his blood, whereas Paul merely awakened after ingesting it.
Leto II and Paul have the conversation with Paul in Children (I think (or maybe it was Leto II talking to someone else later)) where it's pointed out that Paul could have put humanity on the Golden Path but didn't have the chutzpah to give up his humanity in the process. Leto II just saw the inevitability and surrendered to destiny.
Also, late books imply that Paul wasn't the Kwisatz Haderach. Duncan Idaho is.
The Lord of the Rings: "In place of the Dark Lord you will set up a Queen... All shall love me and despair!" Averted, since she refuses the Ring.
Played straight, however, with Sauron and his predecessor and former master, Morgoth. Both of them made themselves out as gods to their orcish and human followers such as the Haradrim. Tolkien himself referred to their reigns as "evil theocracies". Morgoth was somewhat justified, as he was one of the Valar, beings similar to the Greek gods except they did not want to be worshiped, as that was Eru's right to be worshipped alone. Sauron encouraged and enforced worship of Morgoth via human sacrifice on both Númenor, Mordor, and the southern lands.
Morgoth also wanted to be God, to the point he weaved his own soul into Arda at the beginning of time. He claims to Húrin that he actually is God. Húrin calls him on his bullshit.
Morgoth sought to be God because he originally wanted to create. When he learned he could never truly create anything new, his despair turned to destructive rage and hatred for all those who could. As humans are implied to have the ability to create new things of their own (unlike the Valar, Maiar, and Elves), he hates them most of all.
One character's pursuit of this trope is the plot driver for the second Dragonlance trilogy. Raistlin succeeds, with rather horrific results for all involved. Including him. But through a continuation of the somewhat involved time-travel storyline, Caramon warns him of this early enough on for Raistlin to upgrade his condition to sort-of-heroically sacrificed and dead. (Or tortured for all eternity in Hell.) Though this does not prevent him from making postmortem cameos.
Don't forget the Kingpriest of Istar who went as far as demanding the gods serve him. What brought about the cataclysm was his demand that the gods make him a god himself.
Or Fistandantilus, who came up with the whole "become a god" plan that Raistlin hijacked. In fact, he was the man behind the Kingpriest as well. And from The Legend of Huma there was Galan Dracos, whose plan to steal the Dark Queen's powers wasn't as well thought out as the others, though to be fair it took place chronologically first, so they might have been able to learn from his mistakes. Really, this one crops up a lot with Evil Sorcerers in Dragonlance, or any DnD world.
Obould Many-Arrows from Forgotten Realms acquires the moniker Obould-Who-Is-Gruumsh at the height of his power, Gruumsh being the chief god of the Orcs.
Gruumsh sponsored Obould as his Chosen (divine minion given with a shred of godly power without extra strings attached) before this and vassal demigod after (upon death), so it's only a little exaggeration.
And in what has been revealed of the second trilogy, Kellhus gets himself worshipped as one.
In Michael Moorcock's The Chronicles of Corum, the ambitious sorcerer Shool is convinced he has already become a god and is aiming for Supreme God, enlisting Corum for the purpose. In the end it turns out he was merely a puppet of the Chaos Lords, and is left a hollow, dying shell after he has outlived his usefulness.
Inverted in Piers Anthony's Incarnations of Immortality series. Since the original God is too caught up in this trope's attitude to bother with Earthly affairs, another more humble individual must be manipulated into becoming God.
Another interesting point is Satan. According to Archangel Gabriel one of the most effective Incarnations of Evil precisely because Satan does NOT follow this trope, and hence did not become Drunk with Power.
"For though he was master of the world, he was not quite sure what to do next. But he would think of something." - The Star Child in 2001: A Space Odyssey
In Terry Pratchett's Thief of Time, Lobsang/Jeremy becomes the new Time, with control over all aspects of it - although to be honest, it's more of a responsibility and a change of pace than anything having to do with powers. And despite his newfound status, he still submits to Lu-Tze in the dojo, in a brilliant denouement.
Coin, from another of his novels, Sourcery, actually imprisoned all the known gods in a sphere of thought just to prove that he could.
Also, in Pyramids, when now-King Teppic re-enters his home country of Djelibeybi (counterpart to Ancient Egypt), after it has been pushed out of reality, the intense belief of his subjects makes him divine.
Sun Wukong, the Monkey King from Journey to the West. After taking the Ruyi-Jingu-Bang from the palace of the Dragon King of the West, he was given a position in the Jade Emperor's court to satiate his desire for acknowledgement. However, once Wukong realized that his position was a janitorial position, he set up a plot to, and succeeded in, taking over Heaven, declaring himself "The Sage Greater than Heaven." It wasn't until the Jade Emperor asked Buddha himself to do something that Wukong was ousted.
Ironically, after he was freed and assisted Xuangzang in his journey, he DID become worshipped as a god, and in Buddhism is proclaimed the Buddha of Courage (not a godly position entirely, but as close as someone can get to godhood in a religion which itself has no real gods).
This is more or less the plot of Dave Duncan'sGreat Game series, in which all "gods" derive their power from people worshipping them and any dimension traveller can become a "god" in another world. The plot revolves around the attempt by the existing gods to stop a new god from accruing sufficient power to topple them all, and the main character's attempts to stop him by becoming a new god as well.
In Fiona Patton's Tales of the Branion Realm series, the royal family of a fantasy England is physically possessed by a fire god, making the sovereign something of a Christ-figure. Not only does she have divine right, she can prove it. She can set things on fire with a thought, and her family has blazing eyes. Most of the books center on the oddities of religion under such a system: What do you do when God is your mother (and is being abusive)? What do you do when God is a five-year-old child? What do you do when God converts to the worship of a different God? (The answer to the latter is that you seduce God and raise his kid up in the correct religion.)
"And if you are a god, then I now call the Godslayer!"
This trope appears in another main series of David Eddings as well. The main protagonist of Elenium and Tamuli, Sir Sparhawk, is "Anakha", the man without destiny. In the end of the last book it is revealed that instead of just being able to use Bhelliom's powers, he himself is in fact at least as capable as the Bhellion. It's also suggested that it was actually him all along instead of Bhelliom's powers. For comparison, Bhelliom and it's counterpart Klael (which are more or less analogous to the two Destinies appearing in The Belgarion/Malloreon as the Orb and Sardion) have powers beyond any of the deities mentioned in the series.
Also in the Ender’s Game saga, the "God Spoken" on the planet of Path are believed to be spoken to by the gods. This is the reasoning behind their extreme intelligence and the reason they have to "purify" themselves through rituals such as tracing wood grains, counting steps, and being extremely sanitary. In the end they discover that the "God Spoken" have merely been genetically engineered to be smarter, and were also engineered to have an extreme form of Obsessive Compulsive Disorder (OCD) in order to prevent them from being a threat to the government.
In A Nightmare on Elm Street: Protege Freddy tells Jerome that, while he may be limited in the real world, in the dream world he's essentially God. Cue Freddy briefly making himself gigantic and causing the sky to change into a swirling black vortex of thunder and lightning.
Brandon Sanderson loves this trope - all of his major works feature mortals with godlike power who are worshipped as divine. The real gods of his multiverse, though, are called Shards and are incarnate forces of nature (Preservation and Ruin being two named examples), far above anything else in the novels. Some Shards do have mortal intelligences attached to them, however. Book by book, we have:
Elantris: The Elantrians, a race of quasi-immortal magic users worshipped as divine, though it's unclear if they bought into it themselves.
Mistborn: The Original Trilogy: The Lord Ruler, King and God of The Empire, who is actually just a human with a particularly powerful combination of natural abilities. This trope is also invoked by Kelsier who deliberately positions himself as a god to give the masses something else to believe in, so they'll rebel. And in the end of the trilogy, Vin and then Sazed are mortals who become the real deal.
Warbreaker: The Returned are people who died in some notable fashion and then return to life; opinions as to what exactly they are varies across the world, but in Hallandred, where most of the action takes place, they are seen as gods and they (and their priests) run the secular government as well.
The Stormlight Archive: The ten Heralds of the Almighty are apparently (we don't know much about their background yet) humans imbued with a portion of God's power for a specific purpose. During the time the series takes place in, they've mostly passed into legend; Vorinism, the planet's dominant religion, treats them more like angels or saints than gods, but there are other religions that center around the worship of one or more Heralds.
Also Played for Laughs when Sylphrena points out (correctly) that she is a god... or at least a little tiny piece of one. Incidentally, this is true. She is a sliver of the Shard Honor, the closest thing to a God currently extant in the Cosmere.
The Sword of Truth: This may be a little unfair, but Richard Rahl does maintain and encourage the tradition of everyone regularly bowing down and chanting, "Master Rahl guide us... In your light we thrive...Our lives are yours," for four hours a day. Eventually there was a point to keeping this up, but not initially.
Subverted when Conan the Barbarian's girlfriend Muriela, who specialises in impersonating gods, emerges from behind an idol of a goddess glowing purple and giving a different A God Am I speech than the one they had rehearsed. When the worshippers are all off doing her bidding Conan sneaks over and suggests that they leave, at which point she tells him not to be so presumptuous and to clear off before she remembers that he intended to con people in her name. It is strongly implied that the girl lives happily ever after in exchange for periodically renting her physical form to the goddess, who thought the girl was far more fetching than her idol.
Done unintentionally by Leia during the Thrawn Crisis during several Star Wars Expanded Universe novels. After Grand Admiral Thrawn ordered Noghri commandos to kidnap Leia as part of a plot to convert her and her brother, they were at first unsuccessful, but eventually, they recognized her by scent. As it turned out, the Noghri had previously worshipped Darth Vader (Thrawn's predecessor, more or less) as a god after seeing a display of his powers, and after scent-identification made them realize Leia was his daughter, switched their alliance to her and her family, believing her to be divine as well (even so much as giving her the title "Lady Vader".) Leia never actually claimed that she was a goddess, but having the Noghri as allies to the New Republic was too much of a benefit to pass up.
The Big Bad of the New Jedi Order Onimi believes that if he kills everyone in the galaxy he'll get the power he needs to become a god. Somewhat unusually, he also believes that Jacen and Jaina (and several other Jedi) are incarnate gods, and in his mind the whole series is a conflict between the established pantheon and himself trying to usurp them, played out through the Republic and the Yuuzhan Vong. Yes, he's insane. However did you guess?
The Ellimist in Animorphs became a proper god at some point in the distant past.
This seems to happen quite a few times in Dean Koontz's works, with villains in his Frankenstein series, as well as his novel Midnight being the most obvious examples. Even if a villain doesn't believe that he is becoming a god, they are often arrogant in the extreme.
In Dean Koontz's novel Dragon Tears, the villain, Bryan Drackman, is a powerful psychic born with the ability to stop time, create and animate golem bodies in which to stalk his victims, telekinesis, and pyrokinesis. His abilities have grown overtime, and Drackman believes that they will increase enough with practice so he might become the New God and Take Over the World. In order to practice and enhance his abilities, he stalks and toys with the protagonists in the meantime.
At the end of Dean Koontz's short story, "A Darkness in My Soul", a psychic goes on a Journey to the Center of the Mind and finds God trapped in the psyche of an insane genius. He then absorbs God's powers and then takes over the universe after giving half of the power to his girlfriend. Bored with exploring the universe, they decide to start a world war back on Earth for amusement, using humans as playthings.
A classic short story, "Answer", written by Fredric Brown in 1954, has every single computer in the galaxy linked together to answer a single question: "Is there a God?" The computer responds: "There is now."
Played with in The Count of Monte Cristo: Edmond decides that since God hasn't seen fit to reward the good and punish the wicked (quite the opposite, in fact), it is up to man to become God-like. The book does quite a good question of asking what exactly separates Edmond from being a God: he's got enough money to do whatever he wants, intelligence that borders on the omniscience, the willingness and the capacity to destroy or reward those he deems wicked or worthy, a personal gravitas that causes everyone to instantly worship or fear him, he's separated from humanity by both the unfair condemnations of others and personal choice, and, with his mastery of medicine and the legal system, quite objectively holds the power of life and death. The only real difference is that, for him, It's Personal.
Aaron, in Patrick Ness's The Knife of Never Letting Go. He actually believes that he is a saint, and says so in the waterfall-shrouded church right before Viola stabs him through the neck.
In the Star Shards Chronicles, the protagonists struggle with their godlike Star Shard powers, especially Dillon, who is the most powerful. A large part of the second book of the trilogy deals with their "We are gods walking the earth and you should worship us" phase.
Played with and subverted in E.E. Smith's "Tedric" stories. Skandos, a human-like scientist with remarkable technological powers, spends most of his time trying to convince the barbarian Tedric that he is NOT a God. He finally assumes the role when he realises it's pointless trying to talk Tedric out of his belief, and that making an appearance as Tedric's personal deity will have immense benefits to the civilisation of which Tedric is a part, but he himself knows full well that it's only an act. And while Skandos knows he's only mortal (he's murdered three incarnations of himself in parallel universes already), his technological capabilities are so far ahead of the medieval society he's interfering in as to constitute an extension of Clarke's Third Law to effective Godhood.
Neuromancer: At the end the two rivaling AIs Neuromancer and Wintermute amalgamate, and in the words of the new AI it is now the matrix, "the entirety of the system, the whole show". Playing the second variant straight, it tells the protagonist that it has found others of its kind, for example one in Alpha Centauri - and then vanishes.
In one of David Drake's The Lord of the Isles books, Sharina inadvertently travels through time and ends up on an island with a man who claims to be related to the god of storms. His parades are accompanied by thunder created by sheets of tin under the wheels of his chariots.
Aurilelde, from the Young Wizards novel A Wizard of Mars has the kernel of Mars implanted in her and effectively becomes Mars itself, and as she tries to kill Nita she gets angrier and angrier, unintentionally almost ripping Mars apart in her rage.
Jack: If a mere chit can be taken for a goddess, then I shall certainly rule as a god!
Ianira Cassondra is called the living goddess and, thanks to her training under the high priestess at the temple of Artemis in Ephesus, evinces psychic and prophetic powers.
Skeeter Jackson was worshiped as a living god and honorary uncle to the boy who would become Genghis Khan.
In Death series: Chaos In Death has Eve Dallas confront Dr. Chaos, who actually declares "I'm not a man. I am a god!"
In Vernor Vinge's novella, "True Names", when Mr. Slippery and Erythrina forcibly multiplex their consciousness, it gives them the power to take over more machines almost without thinking. Repeat until they (and the Big Bad) have total control over and knowledge of anything connected to any computer ever. The Big Bad detonates several nukes in their silos to make a point in discussion, and it's not a big deal to any of them. At one point, Mr. Slippery is frustrating and rerouting the soldiers sent to kill his real body as a side process while concentrating on something different.
Lindsey Davis' novel Master and God charts the slide into insanity and delusion of the Roman Emperor Domitian, last of the Flavian dynasty and son of the very down to earth and sensible Emperor Vespasian. The sheer burden of responsibility drives Domitian insane and, like other emperors before him, he becomes megalomaniac and insists on his own divinity.
For a heroic example, Arthur after getting five keys in Keys to the Kingdom keeps having to choke back his tendencies in this direction, keeping him from going over the edge.
Nicolae Carpathia in the Left Behind series, particularly after Satan indwells him following his "resurrection".
The first published Horatio Hornblower book, The Happy Return, has Don Julian Alvarado—or rather, "El Supremo." He claims to be a direct descendant of Moctezuma and a god, punishing the "unenlightened" by brutally killing them. Hornblower has to assist his attempt to revolt against the Spanish in Nicaragua, and then has to fight him when the Spanish ally with Britain.
Glory in the Thunder quotes the entire biblical passage in-text. Holders of Aspects of the Divinity are referred to as gods and tend to go mad with power, even though most of them are just as mortal as anyone else, and none are even close to omnipotence.
In Apotheosis this is Adam's bag. He went mad when he learned that The Race, his creators, are extinct, built himself a distributed nanotech body, and went forth to convert all life in the universe to nanoclouds that worship Adam, and kill everyone who refused.
Emperor Ozorne in The Immortals' third book, Emperor Mage. He all but bans worship in Carthak, saying that if the people need to pray, they pray to him. Given that gods in the Tortall Universe are real, and are tetchy, this doesn't go over well.
Bleu, the muse of Sacré Bleu, once refers to herself as "a fucking goddess", and is not at all unjustified in doing so.