Risen Martyr — "Hey look, I died, and now I can do all this awesome shit!" That's right, folks, D&D includes a prestige class that makes you Jesus. Subverted in that the powers gained are pretty modest, and it doesn't say much about a person if dying is a positive career move.
Increasing in level in any class arguably makes one more of a badass (that 1st level wizard having troubles fighting kobolds? He's instantly killing Pit Fiends now). It's just that the badassness is split out over twenty levels, which generally makes the shift somewhat gradual.
Incantatrix, Planar Shepherd (which deserves special mention for being the only prestige class that's strictly better than just taking more Druid levels), Dweomerkeeper, Ruby Knight Vindicator, Jade Phoenix Mage, Survivor... There are too many to list, but here's a discussion of particularly badass levels.
If none of the official classes are badass enough for you, try these fan-made rules.
4th edition D&D goes so far as to build this into the classes twice over. First, at level 11, you get a Paragon Path that lets you take a level in badass... then, at level 21, you get an Epic Destiny, which is a second level in even more badass. Epic Destinies include the likes of Master of Moments (you gain control over time), Undying Warrior (there is nothing in the multiverse that can kill you), Parable (you realize that all of reality is but a story... and you ascend to become its author), World Tree Guardian (you are responsible for protecting the spirit of the world itself from harm), Chosen (your god makes you it's avatar within the world), and Demigod (you become a lesser god).
One famous example in Dungeons & Dragons cosmology is the case of Glasya, the daughter of Asmodeus, the Overlord of Hell. Originally portrayed as a glorified erinyes with the attitude of a rebellious teenager (and, at times, a Spoiled Brat), sometime during the 3.5 Edition she came to an understanding with her father, was granted a great deal more power, and gained the rulership of the sixth layer of Hell. Now, she's as powerful and cruel as any other ruler of Hell, and truly is her father's child...
Most any RPG character will level up through the course of the game.
If you're playing Deadlands, death can be step one to taking that level. Just ask Abe Lincoln who came back from the dead to run the Union's monster-hunting agency.
Warhammer 40K - the original Space Marines took a Level in Badass and a Level in Kindness. When they started in Rogue Trader, the Space Marines were said to be the mightiest warriors in the Imperium and much feared throughout the galaxy as unstoppable, cannibalistic, psychopathic savages in graffiti-ridden armor with pointy noses. In reality, they had an underwhelming Toughness of 3! That meant they were no tougher than your average human or eldar or whatnot. Their "Beaky" power armour was also weaker, it was 4+ compared to the current 3+ (though you could buff it to a 3 if your Space Marine found some Flak armour to put under it back then). Also every character had to roll to see what equipment they received, if I remember correctly - your Space Marine could end up packing a measly lasgun or autopistol if he rolled low enough. Finally Space Marine stats topped out at 6 in everything, the same level of potential any ordinary human could achieve in a career of almost the same amount of time and Marines would lose out in comparison with quite a lot of aliens. Now Warhammer 40k has made Space Marines distinctly superhuman in stats (especially in the RPG-oriented games) with alot better equipment, not mention that they're nobler, more disciplined and have gold-trimmed armor with more intimidating helms.
The Imperial Guard took a whole bunch of levels in bad ass this year after they got a new codex that's actually good. Also an example of Throw the Dog a Bone since IG and IG players were Games Workshop's favorite Butt Monkey.
Space Marines also took a few levels in badass with their Fifth Edition Codex, the most notable being the Captain's Weapon Skill boost and the Chapter Master's Call Orbital Bombardment ability.
The Eldar got a significant boost with the rules changes in the Fifth Edition rulebook that combine with existing powers to make their Falcons tougher than Land Raiders.
The recent wave of updates to extremely old codexes (most notably the Dark Eldar, who have not received a new book in over a decade) resulted in many taking several levels in badass. Most notably the Grey Knights, going from an extremely small and elite group of special forces space marine taking massive death tolls to banish even the smallest of greater daemons to badass demonhunters whose very existence can make demons cower (and their leader carved his name into the chest of a daemon primarch). Some of these were considered well deserved and long overdue (notably the Dark Eldar, which was praised as being both flavorful and balanced) others were not.
In-Universe, the Tau. When first found by the Imperium, they were plains-dwelling stone-age hunters. Imperial standard doctrine is to eliminate any native Xenos species to prepare the planet for human settlement, but a huge warp storm destroyed the fleet and cut off T'au from Imperial contact for 6000 years. When the warp storm cleared, the Tau were found to be a highly-advanced spacefaring species with a small empire. The Exterminatus order still stands, but as for actually doing it...
Most kinds of Exaltation are the epitome of taking a level in Badass, taking a mortal with promise and making them a powerful demigod. There are exceptions, but in those cases (Such as Alchemicals and Liminals) the Exalt comes into existence as an Exalt. Many mix it with something else, though:
The Solars took a collective level in Badass when the Jade Prison broke. For better than 15 centuries, spanning the Shogunate, through the Scarlet Empress's reign, there were roughly twenty Solar Exaltations active in the world at any given time. 2 years after the Scarlet Empress disappeared, that number shot up to 150. All of a sudden, the idea of a Solar Resurgence isn't that crazy, even in the face of the Wyld Hunt.
The Free Council in Mage: The Awakening. They began as the Nameless, a disorganized bunch of people who didn't fit in with the established Atlantean Orders. Since they were the Butt Monkeys of magical society, the Seers of the Throne tried to recruit them with the promise of Better Living Through Evil. The outcome was a lot of dead Seers and the creation of the Free Council as a formal order, and ever since, the Council built up a reputation for innovation and really coolMagitek.
One of the oldest tabletop games in the world: Chess. The pawns, if they reach the other side of the board, may become promoted into any piece, most often a queen (sometimes a knight, if need be)