Mesopotamian Mythology: Both Gilgamesh and Enkidu from the Epic Of Gilgamesh. In fact, they may be the very first recorded badasses ever. Their greatest deeds are the killing of the giant Humbaba and the Sky Bull, a rampaging monster that was sent by the Gods to Earth as a punishment.
The god of desert and storms, Seth, even if Hijacked by Jesus in modern times, in real Egyptian myth was Ra's top enforcer and warrior. Guess who kicks the Demon Serpent Apophis's ass every night and ensures the the sun can rise again? Also, Horus once challenged Seth to a race on stone ships... Seth built his one by carving a mountain with his mace.
In one myth, Mankind grew disrespectful of the gods, and the gods sent one of their own to teach them a lesson. Who did they sent? The bloodthirty and lion-headed goddess of war Sekhmet? The all-mighty Seth? Nope: the Eye of Ra, Hathor. Who, in one day, massacred more than half of mankind, and would have finished the job the following day had Sekhmet not seen the sea of blood from Hathor's extemination and got the other gods to create a lake of beer to get her drunk.
Chinese Mythology: The Eight Immortals who, according to legend, founded a kung fu style, had a number of stories about how Bad Ass they were. For example, one got so drunk that he passed out and got mauled by a tiger for a while before waking up and punching the tiger to death. Or the time when the Crown Prince of the Sea kidnapped their token female member and they defeated his entire army, plus his elite guard, and almost destroyed the palace.
The Monkey King, who is pretty much a professional monster-fighter that came very, very close to completely wrecking Heaven, and kicked the ass of several major gods.
Heracles is tough enough that the gods kick him off the Argo because the Quest of the Argonauts, the epic adventure of the age, apparently wouldn't be challenging enough for him. Instead he goes off to perform his Twelve Labors, during which he kills an invulnerable monster, holds up the sky for a while, frees Prometheus, steals the guard dog from Hades, and mucks out some stables... by redirecting the course of a river all by himself. As Heracles was so strong, a Rasputinian Death was needed to do him in: When he donned a poisoned garment, the poison began to tear his body apart, rending flesh from bone, but did not kill him. The pain was incredible but the death was a long way off so Heracles burnt himself to death on a pyre - which he personally built by tearing down full-grown trees while suffering from the poison.
Typhon, the original Hero Killer, who storms Mount Olympus by himself, makes the gods run in fear, and defeats Zeus the first time they clash. Zeus is one as well, for having the nerve to stand up to Typhon in the first place, and beating him the second time. Also Selene, who according to Nonnus beat the crap out of Typhon, and still has the marks of that conflict, the lunar craters.
Athena is the reigning Queen of Badass in Greek mythology. In some version of the Typhon myth, even Zeus runs away, but Athena doesn't...and then she taunts him about it, while facing down Typhon, until Zeus relents and takes up the fight himself. The Iliad is also, among other things, an extended series of examples of why messing with Athena is a really, really bad idea; every time she appears on the battlefield it ends with a rout for the Greeks (her chosen side). Among her accomplishments are defeating Ares with Diomedes and disarming Hector to allow Achilles to kill him. While most myths centered around her emphasize her wisdom, she is also the Goddess of strategic warfare, and she routinely out-badasses the other God of War, Ares. It is very wise to remember that, had she been born a son, it was prophesied that she would have overthrown Zeus.
Almost every character with a major role and who is not Paris or Thersites in The Iliad. On the Greek side, there is Achilles, Agamemnon (just read his rampage in Book 11), Ajax, Diomedes, Odysseus, Idomeneus and Meriones; on the Trojan side, there is Hector, Sarpedon, Glaucus, and Aeneas (who fights both Diomedes and Achilles and survives).
From The Iliad: Diomedes, despite not being half-divine like most Greek heroes, wounded two gods in a day: Athena drove his chariot while he speared Ares, chasing the god of war off the battlefield. Then, he threw a rock which two ordinary men couldn't even lift at Aeneas, the son of Aphrodite, and when Aphrodite appeared to heal Aeneas, he wounded her in the arm, which sent her whimpering off to her daddy Zeus. And he survived all of this!
The Aeneid focuses on the heroics of Aeneas, but Turnus and many soldiers on both sides of the Trojan-Latin war establish their ass-kicking credentials as well. The stand-out may be Mezentius, an exiled Etruscan king and notorious god-insulter.
Let's face it: if you're an hero in Classical Mythology you are a badass. To prove the point, let's examine Jason, possibly the weakest one: one of his tasks to retrieve the Golden Fleece included wrestling into submission two gigantic fire-breathing bulls of bronze made by Aephestus and yoke them, and the evidence he's less badass than other is that he needed Medea to give him a fireproof salve before overpowering the gigantic bulls of bronze. Yeah, the guy who can overpower two gigantic bulls of bronze made by the God of the Forge at the same time is considered a pansy...
Some of Cuchulainn's feats defy logic. In Táin Bó Cúailnge, Cuchulain alone did enough damage on the army of Connacht marching upon Ulster that he seriously slowed them down for days, giving the warriors of Ulster enough time to assemble their army. In his death throes, he tied himself to a rock to remain upright, and his enemies refused to believe he was dead until a bird landed on him.
Fionn mac Cumhaill was a kind of Irish King Arthur who headed the Fianna, a Badass Army dedicated to defending Ireland from invaders. In The Battle of the White Strand, all the kings of the world gang up on Ireland — and lose.
Beowulf: Beowulf tears Grendel's arm off with his bare hands. If we believe his boasting, he also swam for five days in chainmail and carrying a sword, and killed nine sea monsters during the last night. When he was already an old man, he took on a dragon with only one warrior helping him.
In Aztec Mythology you have Huitzilopochtli, who was born fully grown and clad in armor just in time to save his mother from her older children (all of them powerful gods) who wanted to kill her, cutting them all into little pieces with a weapon made of fire/lightning. He then went on to become the patron of warriors, chief god of the Aztecs and the sun itself.
In Mayan mythology told about in Popol Vuh, twin heroes Hunahpu and Xbalanque fight evil Underworld gods and super-powerful bird demons. They are so badass that somehow they can survive being decapitated by giant man-eating bats and re-attach their limbs when ripped off by enemies.
Veborg, a viking shieldmaiden, fought in the Battle of Bravellir. After killing a king's champion, proceeded to cut the chin of Starkad, a part-giant who is generally depicted as being at least twice as tall as anyone else. She was finally killed by Thorkell the Stubborn after "many wounds and much verbal arguing".
Speaking of Starkad, not only did he lose his chin to Veborg, he received one wound so large that one lung was hanging out, had a massive cut in his skull and lost a finger on his right hand. He survived, which is more than can be said for any of the many men that Starkad fought that day. Even when he was very old and virtually blind, Starkad killed several men, including some who were mounted on horse and charging him.