Sometimes playing as a mere mortal just isn't awesome enough. Some games are content to give the player godlike power over their worlds, or a nigh-omniscient perspective, but others make no bones about it and say "A God Is You!"
This is a gaming trope that comes in two flavors:
Flavor A - The main character of the game is a god or powerful spirit, facing down godly threats, on a quest to reclaim their power or leading their civilization to glory.
Flavor B - The game breaks the fourth wall by casting you the player as a (or the) god.
Nobilis, where you play as a mortal that has been raised up to be an Anthropomorphic Personification of any element of the world, from Dreams to Water to Computers. From the start of the game any Noble has enough power to destroy the world or change great parts of it, and you only get stronger. Most of the conflict of the game tends to revolve around using social manuevering, politicking, and in general not using your whole power to flatten everything in the way, because when your opponents also have enough power to blow up the world, it pays to play nice and be indirect.
Scion, where the player characters are "merely" children of the gods to start with, but can eventually become a mighty pantheon.
Similarly, divinity is one of the possible "Epic Destinies" for characters in Fourth Edition Dungeons & Dragons.
Though you don't get to actually play the character as a god (or, at least, there aren't any rules for it).
An older edition had the "Immortals Rules", which are exactly that, although the word "god" was never uttered to avoid the wrath of the Moral Guardians.
Deities and Demigods, provided two things: stats for the D&D pantheon and various historical pantheons (Greek, Egyptian, Norse), and rules for building your own deities. Along with suggestions for how to get your PC party into godhood/keep the game running afterwards.
Amber Diceless, which is based on The Chronicles of Amber novels. Basic PCs come in two flavors (Princes of Amber and Lords of Chaos), each of which can use their special power (the Pattern and the Logrus, respectively) to essentially create Alternate Dimensions at their pleasure and shape and outfit them how they choose. The corebook notes repeatedly that spending creation points on personalized weapons, servitor creatures, and even private dimensions for your character is a luxury (it ensures that the character will always be guaranteed access to them), and that the characters can just create or find whatever they want for themselves once the game actually begins.
In virtually any superhero game that allows custom characters, you could theoretically play any member of any pantheon in the world. Case in point, when playing with stock characters, you can play Thor or Hercules just for starters. And in DC, you could be Orion or Lightray if you want.
The game of The Darkness might qualify, since you're a vessel for the titular Darkness, the manifestation of the very soul of evil. So you're a DARK god, but still a god... And no, Dark Is Not Evil is not present.
God of War: Specifically the second, which takes place after the hero has overthrown Ares and taken his place. He ends up having to regain his lost power, however.
Godzilla Unleashed lets you play as Mothra (who is worshiped as a goddess on her home island).
Likewise, the Wii version of the game allows you to play as King Seesar. A guardian god-like monster that's loosely based off of the Shisa of Okinawa folklore.
Both versions of the game allow you to play as King Ghidorah and Baragon as well. Mothra, King Ghidorah, and Baragon were gods/guardian monsters in the film ''GMK.
To say nothing of GOD-zilla himself. And Megalon was the god of the Seatopians...
For that matter, you can play as Mothra in both NES Godzilla games, and most Godzilla games in general.
Ōkami casts you as Amaterasu, the Shinto goddess of the sun. Not only do you kill monsters and fight an Eldritch Abomination, you also make plants come back to life and grow, and answer the smaller and larger prayers of all kinds of people you meet on your way. Most don't suspect that a white dog has anything to do with it, but hearing them praising the sun and making offerings to show their gratitude really makes you feel good.
Populous, an example of the "leading your civilization to glory" variant. This might more properly be a case of flavor B, though.
In Summoner 2, you are the Child of Prophecy, the Divine Queen, and about as close to being a goddess as you can get without living in Heaven. You rule a kingdom, have your own temple, and people build huge statues in your honor. You also have some wicked powers, of course...
Heck, the ORIGINAL Summoner revealed that you're 1/9th of a dead god.
In Terranigma, "Light Gaia" and "Dark Gaia" are powerful other forces that will eventually be corrupted into being called "God" and "Satan", but nonetheless, Ark is told that he is what mortals would call a god.
Every "major" character in Valkyrie Profile either starts out as a god of some kind, or becomes one when Lenneth Valkyrie picks them up to be her einherjar (they even get their goodhood ranked), except for the character Celia, which is by some regarded as having the saddest storyline because of the fact that she doesn't die, everyone else does and she's left alone.
There's a subtle distinction made to keep the game interesting: you play as the beings whose permanent roles are to be the Horsemen, not with the invincible power of the Horsemen themselves. While this still puts you in this trope's territory, that you're always acting "outside of your office" means it's possible to actually be challenged or to lose.
From Dust, which borrows heavily from Polynesian mythology, casts you as "The Breath", the guardian spirit of an island tribe who uses its ability to manipulate land and water to help the tribe recover its lost history.
Doshin The Giant casts the player as a sun god who dies each evening, but is reincarnated the next morning.
God of Thunder: You play as Thor. He doesn't seem to have particularly godlike powers, though.
In Xenoblade, Shulk becomes one in the ending, although he throws away his power in favor of recreating a world without gods.
Reus, much like SimEarth, also casts you as a Gaea-like entity - albeit in a much more stylized fashion. Rather than acting directly, however, you alter your world through 4 tools known as The Giants - each a gargantuan being larger than a mountain. Commanding the Giants, you can alter the planet's surface on a grand scale; digging oceans and raising up mountains, or making fertile woodlands spring up from the ground; or on a smaller one - creating lush orchards and packs of wild beasts for the emerging humans to hunt, or pockets of valuable minerals and materials for them to mine, etc. Interestingly, you have no direct influence over the path taken by the human tribes that spring up on your surface - you can only assist them in whatever pursuits they choose to favor, or destroy them if they displease you. If you play your cards right, they'll worship the Giants as gods and revere them. If you don't... well, turns out even mountain-sized giants aren't ENTIRELY invulnerable, if enough humans pile on them...
Magic: The Gathering features the tagline "You Are A Planeswalker". Planeswalkers are the closest you can get to godhood in the MTG universe: with enough study, a planeswalker can do just about anything short of creating life.
In the Theros block, we even have an example of a planeswalker becoming a god.
You have a sort of character in Black & White, but you never see it and supplicants address the screen directly. The other gods you see are points of light with a hand.
Super Paper Mario uses this as a way to lampshade the gameplay instructions. Mario doesn't know what all of this "Press A" business is, but the other characters assure him that the great being that watches over them all understands.
The Sims Medieval makes this explicit by having the player fill the role of "The Watcher," a deity that made the land and gently guides the hero characters to making either right or wrong decisions. There are even two religions based on you: Jacoban and Peteran. Jacobans believe that The Watcher is cruel, while Peterans believe that The Watcher is benevolent.
In Patapon, you're the god of the Patapons, called the Almighty.
In the Civilization games, you're cast as the ruler of a civilization, but you stick around for however many millennia you feel like playing, remain in power through any and all revolutions, and have the power to manipulate any of your cities without having to route through whatever system of government you have in place at the moment. Democracy? Bureaucracy? Theocracy? Feudalism? This micromanaging god cares not.
It's possible that you're not playing as that particular nation's leader per se, but rather the nation itself.
Baten Kaitos has the player as a "Guardian Spirit" guiding the protagonists.
The sequel uses the same principle. Before actually pulling a Tomato in the Mirror on the player, revealing you as part of the God of Darkness.
The merest subtle nod to this in Baldur's Gate: one of Jaheira's selection quotes is "Yes, oh omnipresent authority figure?"
At the extreme opposite is the insane Tiax with: "One day, Tiax will point and click."
Edwin has one too: "I do not understand this 'mouse magic' that makes me do your bidding."
A minor character in Beregost exclaims "Don't click me! I don't want any trouble!"
A few other quotes subtly nod to the mouse interface too. Minsc has "You point, I punch!", and Anomen has "Point the sword and I shall strike!" (The cursor is a sword when hovering over a hostile unit or when a weapon is selected.)
In Ever17, it is all but stated that you are Blick Winkel, the 4th dimensional being who helps the protagonists at the end. While not technically a god, you are still a being of a higher dimension than the protagonists who can travel through time.
Made even more obvious by the fact that "Blickwinkel" is the German word for "perspective".
In MARDEK RPG 3, when complimented by the king at one point on a job well done, Mardek smilingly replies, "I can only move at the will of my unseen master!
Implied in the first series of Dawn of War games, as the units (barring the Necrons) seem to talk to the player as if they're even higher-ranked than their commanders, or if the commanders themselves are referring to someone higher ranked than them. To really drive it home, weak Chaos units will ask the player to bless them, although other units will sometimes talkback to the user.
Dawn Of War 2 more or less confirms this, since your units in campaign mode are apparently responding to orders from your Heroic Mime Force Commander. Played straighter in Retribution, where all units go back to referring to the player as their CO.
Dwarf Fortress Fortress Mode — popular speculation is that the player is Armok, God of Blood.
Another popular theory is the exact opposite — no, player, you are the Hidden Fun Stuff.
The "Virtual Villagers" series of games has always done this to a certain extent, with villagers engaging in festivals to honor the "Guiding Hand," a reference to the hand-shaped cursor. The fifth game, "New Believers," takes this one step further, giving the player godlike powers that they earn by building their "god points"
While it's implied in other lines the player is a king or a commander, one of the "pissed" lines of the Human Knight in WarCraft III is "I have been chosen by the big metal hand in the sky!". Clearly an allusion to the cursor for the human faction being a gauntlet.
The Mother games have the characters thank the player for their assistance towards the end of the game.