Beowulf tore Grendel's arm off with his bare hands, and later wields a sword made by (and for) the giants. And then he broke the sword by swinging it too hard. 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.
And when he was an old man, he killed a fragging dragon with only one other person helping him (even if he died too), so he qualifies as a Badass Grandpa as well.
Cuchulainn, the Irish mythological hero/pimp is an all around Badass. Some of his feats defy logic and he has examples of an Unstoppable Rage. Seen 300? Where the 300 guys fight millions of guys, and they lose? When Cuchulainn did that sort of thing, he won. And he didn't have 299 guys behind him. That was all Cuchulainn, baby.
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.
And then, after he was dead, he cut of the hand of the guy who killed him.
And before Cuchulainn, there was Fionn mac Cumhaill. Ireland's version of King Arthur, complete with his own band of Bad Asses, the Fianna.
Sticking with the Celts, how about the Arthur himself? He, Cai, Bedwyr and the rest of his Badass Crew spend most of their time inconveniencing witches, giants and massive lions by killing them. Because that's how they roll.
In one of their clashes, Horus Challenged Seth to a race on stone ships...Seth built his one by Carving a MOUNTAIN with his mace.
Hathor was another badass deity, the only action goddess able to kick Chaos demons's asses, who takes a level in badass after have butchered thousands people as punishment for a conspiration against Ra (in reality, she already punished the conspirator, the carnage she did was for the evulz).
Coming in from China, the Eight Immortals who, according to legend, founded a kung fu style had a number of stories about how Badass 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.
Heracles from Greece is tough enough that the gods kick him off Argo because 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.
With a tree trunk, no less!
The way I knew, it it was a boulder.
His death was equally badass and fits under Rasputinian Death. He wore a garment that had been smeared with blood containing the poison of the hydra; the poison began to tear his body apart, rending flesh from bone... and he survived that. The pain was incredible but the death was a long way off so he had himself burnt to death on a pyre... that he personally built by tearing down full-grown trees while suffering from the poison.
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 attacked her too. And he survived all of this!
Achilleus is the greatest. The guy was an unstoppable warrior (though he had one weak point) and you can't even say his name without "kill"! Every character in the Iliad who isn't Paris or Thersites gets Badass points. We have Hector, Achilles rival and Badass Normal, Agamemnon the (despite what movie adaptations will tell you) the patron saint of Authority Equals Asskicking (just read his rampage in Book 11), Ajax, the original Big Guy, Sarpedon, who's essentially the Trojan response to him, Sarpedon's friend Glaucus, Odysseus, one of the first Badass Bookworms, Idomeneus and Meriones who team up with Poseidon to keep Hector away from the ships, Aeneas, who kills lots of Greeks, then fights first Diomedes, then Achilles and survives, and Nestor, who's a Retired Badass. Even Patroclus manages to be pretty Badass, what with the whole killing Sarpedon thing. Nestor was also one of the Argonauts back in the day.
The Aeneid continues the Badass-ery, with Aeneas, Turnus and many soldiers on both sides of the Trojan/Latin war establishing their ass-kicking credentials. The stand-out may be Mezentius, exiled Etruscan king and professional god-insulter.
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 (or lightning, say some). He then went on to become the patron of warriors, chief god of the Aztec AND the friggin sun itself.
In Mayan mythology, twin heroes Hunahpu and Xbalanque have fighting evil Underworld gods and super-powerful bird demons as their favorite hobby. 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.
Typhon, the original Hero-Killer, who storms 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.
Veborg, a viking shieldmaiden who, 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.