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.
Jason; leader of the Argonauts, was completely human and was not descended from the gods.
Theseus, legendary slayer of the Minotaur, may have been without any divine heritage (although some sources say that he was a son of Poseidon), and thus a Badass Normal to be reckoned with. He also won over Hippolyta, queen over the Amazones in single combat and married her.
He and Heracles also literally Walk into Mordor (well, Hades, if you want to be pedantic).
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 impressed Athena, the goddess of cunning herself, with his brains and wit.
Atalanta. Unusual 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!
Diomedes, This guy had not a drop of godly blood in his bones, and he is considered the second greatest greek warrior during the Trojan War (when you considered that the Greatest is a demigod that is saying a lot). Manage to defeat Ajax in a wrestling match and defeat both Hector and Aeneas. He is considered as intelligent as Odysseus (his name actually means God-like Cunning). He also manage to INJURE not one but TWO gods, and survive an encounter with a third. Also he is one of the few who manage to make it home alive (though he was later banished by his cheating wife). And eventually sailed to Italy and founded several cities.
From Norse Myth, we have Sigurd, who had invulnerability after bathing in dragon's blood (complete with the single vulnerable spot, on his back). But he still had to singlehandedly slay that dragon.
Also from Norse Myth is Thjalfi, a farmer's son who became Thor's servant. He slew the nine leagues wide giant Mokkerkalfe by himself, and just barely lost a race against the Anthropomorphic Personification of Thought.
Beowulf, from the ancient English epic Beowulf, a nordic hero who slays monsters and becomes king without any divine aid or even mythic weapons. He pretends to be asleep when the monster Grendel attacks the human village who called for his aid so he can surprise it, and decides to rip the giant's arm off with his bare hands. He goes on to kill its mother and then later a dragon, before succumbing to his wounds in his last fight.