Badass Normal/Oral Tradition

Ajax from The Iliad. All of the other mighty warriors have either gods helping them or are nearly invincible to begin with. Not only does he never get help from any of the gods, he also singlehandedly holds off the Trojan army during their assault on the Greek ships. He also fights Hector to a draw, twice.
Patroclus' badassery is sadly under-appreciated. Not only is he completely mortal and not aided by any gods, but when he leads the Myrmidons into battle on Achilles' behalf, he beats the Trojans back from the ships, across the beach, and all the way to the walls of Troy, slaying several god-assisted heroes on the way, including Sarpedon, who is the son of goddamn Zeus (though that isn't an incredibly difficult achievement). And then, when he tries to take Troy itself, it takes the physical intervention of Apollo to slow him down long enough for two -- two! -- Trojan heroes to finally kill him. He even tells Hector, just before the man delivers the killing blow, "You are only my third slayer." And you thought he was just Achilles' boyfriend.
Achilles's father. Because any son that Thetis bore would be stronger than his father, Zeus set her up with a rather weak mortal. He held onto Thetis, and in order to shake him off, she changed into several hideous forms but he continued to hold on through sheer determination. Thetis was so impressed with this she consented to marrying him - and Achilles certainly became much stronger than his father!
Odysseus. Probably one of the best known Greek heroes after Heracles and Achilles, and unlike them, he wasn't the son of a divine parent (although he was the great-grandson of Hermes). Came up with the idea of the Trojan Horse, survived the wrath of Zeus, Poseidon and Helios (the sky, the sea and the sun respectivly) using just his wits and luck, and even impressed Athena, the goddess of cunning herself, with his brains and wit.
Atalanta. Usual for an Ancient Greek hero in that she has no divine parent and also the fact that she's a girl. Left to die because her father wanted a son, she survived, grew up in the wilderness raised by a bear, she was attacked by two centaurs at the same time, and killed them both. She later joined a group of heroes (most of them the sons of gods) in the hunt of the Calydonian boar, a giant monster sent by Artemis and was the first to wound it. When she was reunited with her father, he told her she had to marry. She agreed, but would only marry a man who beat her at a footrace. She was beaten, but her opponent, Melanion, had to get help from Aphrodite, the Goddess of Love. According to some sources, she was even one of the Argonauts!