Marvel Comics' series Nick Fury's Howling Commandos (not the one set in WWII). Starring the likes of Warwolf, N'Kantu the Living Mummy, Frankenstein (a clone of the original Frankenstein's monster), Gorilla-Man, and a zombie named John Doe.
Along similar lines and years earlier, DC Comics' Creature Commandos. Unlike the Howling Commandos, who were taken from a pool of supernatural creatures known to SHIELD, the original Creature Commandos were all normal humans at one point, deliberately transformed into monsters by scientific means for the purposes of psychological warfare.
Dungeon Monstres #1 gives us the merry band of Juan-Juan (A Bogeyman fighter), Darmfloor (An Undead magician), Wilfried (An Elephant barbarian-monk), Yomanda (An Ochtone with no discernible class) and... Willem van Dattum (An evil salesman who bears the Sword of Destiny at the time).
The War of the Spider Queen series is mostly about a bunch of drow on a quest to find out why their boss doesn't answer calls. They even took a Draegloth with them, and later sort of picked up an Alu-fiend. Of course, unlike Salvatore's most famous Drow protagonist, they're all evil.
Dungeons & Dragons always was ready to throw an occasional monster at least as a sidekick, eventually including just about everything.
The Basic D&D supplement "Gazetteer 10 The Orcs of Thar'' had rules for creating and playing humanoid monsters such as orcs, trolls, hobgoblins and so on.
AD&D2 has "The Complete Book of Humanoids" including traditional Mook species.
In D&D3 this became a routine procedure. As mentioned in The Order of the Stick example, there are rules for monster NPCs, but most of the monster races aren't really that good as player characters. The reason for that is level adjustment. It means that to pay for the increased power of, say, being able to eat your opponents' brains and mindblast them, some of your levels don't actually count for the character's advancement, meaning you're perpetually between 2 (drow) and 6 (mindflayer) levels behind every other player character. While the abilities you gain are pretty cool, they're often no match for the increased Hit Points and especially spellcasting power actual levels would give you.
In 4e, with the NPC stats at the back of the first two Monster Manuals, playing a monster PC is more mechanically viable than ever.
One D&D book (containing ready-to-use NPCs for various occasions) contained such a group of adventurers: a troll fighter (full plate but fights with claws), an ettercap cleric (rather smart for an ettercap but Int isn't his casting stat anyway), a pseudodragon sorcerer (ditto), a phase spider rogue (who needs lockpicks when you can pass through walls?), and an umber hulk monk (who fights blindfolded, purely by tremorsense, to protect his teammates from his gaze). A Ragtag Bunch of Misfits if ever there was one, but rather friendly if you don't attack right away.
Forgotten Realms from AD&D1 times had Mindulgulph mercenary company, mostly non-humanoid. Named after a ruined castle that one warrior lady turned into her base mainly through use of telepathic communication first on everything moving during a dungeon crawl.
The Dark Sun setting includes the Thri-Kreen — anthropophagous mantis warriors — as player characters from the get-go, with further expansions providing at least six different Kreen sub-species.
Pathfinder overtly tries to avert this trope, emphasizing the Always Chaotic Evil nature of its monsters, but at the same time includes a wide variety of monster races, and much more balanced than those in its ancestor, D&D.
Neverwinter Nights. The Badlands servers allow you to choose from more than 60 monster types to use as your character's race.
Among the many playable races in Dungeon Crawl are kobolds, mummies, ghouls, and orcs, the last of which can even get other orcs to follow him/her as their messiah.
Disgaea offers your entire cast as demons or angels, and you can get every single monster species you find and equip them with weapons AND armor. Then again, it's Hell.
World of Warcraft: Horde player characters are all about this, with the Horde representing a goodish-neutral version of the Fantasy Axis of Evil — orcs, trolls, undead, "minotaurs" (Tauren). The expansions mix up this pattern when adding new races to both sides, with for example werewolves (Worgen) going on the Alliance side.
The Charr of Guild Wars 2 are this to those who played the original Guild Wars, where they were evil and man-eating (which was actually propaganda); after dethroning the zealots using a false religion to guide them, they're now in an uneasy truce with the humans to fight their common foe, the dragons, and wander the world to that end.
In the second game of the SaGa series, your entire party (Main character included) can be monsters. The third game starts all of your characters off as human, but allows them to become monsters.
The sample game that came with the original Playstation RPG Maker tool starts with an A.I. controlled party slaughtering goblins. After respawning, Gobli decides that he'd rather be an adventurer than XP fodder, and sets out on a quest to do so. He is joined by a literal hole in the wall that wants to become an NPC and a kitten that wants to be a Final Boss.
The Dragonborn in The Elder Scrolls V: Skyrim is a somewhat example, being a Dragon in human/oid form. Dragons in this setting are innately driven to destroy and dominate others, something that even the reformed dragon Paarthurnax admits he struggles with. He notes that with their kinship, those same urges are present in the Dragonborn as well.
Dark One: You made us and the humans, but they get to be PC races while we are slaughtered by adventurers? Marduk: Well, there are those rules for monster PCs— Dark One: Those rules are crap and you know it!
Later Malack turns out to be a vampire, which has a punishing level adjustment of eight, was previously part of an adventuring party, and notes that the level adjustment suffered in a Dungeons & Dragons world does make it difficult to find "appropriate" challenges to level up on.
Xykon, as a lich, has a level adjustment of four, although it doesn't seem to impede him nearly as much as the one above despite being into epic levels even before the adjustment is calculated. As the primary villain of the comic he may benefit from accelerated xp growth to keep him outmatching the heroes though.
Yet Another Fantasy Gamer Comic plays with it. Just look at its article's page picture. It started with beholders, kobolds and harpies, continued with a lich and drow, and most of the time retains at least a token mongrelman even when it mainly involves more traditional PC races. Sometimes Always Chaotic Evil aren't very chaotic or evil, sometimes they very much are, and sometimes they have a reason to team up with "good" guys.
The eponymous anti-hero party of anti-HEROES is composed of a vampire, a ghost, a tiefling, and a half-demonic (or is it half-divine?) mysterious cloaked figure. They are also much more sympathetic than the nominally "hero" party including more standard races.
By The Book is about a party consisting of a goblin, an orc, and a kobold. Later they're joined by some humans and a tiefling. And their "former" employer sent a group consisting of a couple goblins, a hobgoblin, a half-orc-half-ogre and a gnoll to retrieve them, who have gotten somewhat sidetracked by a series of unstable portals.
Unsounded: Duane may have been blackmailed into guarding Sette as they trek across the continent at first, but he seems to be genuinely enjoying it.
Dan from Dan and Mab's Furry Adventures is an incubus and retired adventurer; however, he was never both at the same time. His cubi-based powers didn't manifest until after he quit adventuring, so during that time, he was just an adventurer with wings; and is now an incubus with adventuring skills.