Murtagh sets the stage in the first book when, with only his bow, he killsdisembodies Durza, after said sorcerer defeated Eragon almost effortlessly.
At the end of the first book, Eragon is outmatched in a fight with Durza in the main chamber of Tronjheim, surrounded by urgals, while Saphira and Arya are stuck in the Dragonhold a mile above his head, separated from him by the seemingly indestructible Star Sapphire, a sixty-foot wide and thick gemstone that is the heart of the dwarves' city. Right when Eragon is injured and a moment away from defeat, Arya uses her magic to break the Star Saphhire, shattering it, while she and Saphira fly down through the falling fragments toward him, Saphira breathing fire for the first time. Durza is momentarily shocked and stares, distracted, and in that one brief moment, Eragon summons up the last of his strength, sets his sword on fire, and stabs him through the heart, utterly destroying the Shade.
In the second book, Eldest, Roran (a minor character from the first book, Eragon,) proves his awesomeness by not only rallying the village into taking up arms against a squad of trained military soldiers and a pair of human-like condors who eat flesh, but afterwards convinces nearly the entire village to abandon their home, march over freezing mountains, steal a cargo ship from a guarded city, sail through a whirlpool, and join a rebel movement. This troper sometimes opens the book just to read his parts of the story!
In Eldest, Murtagh earned the respect of many when he announced his return by kicking Eragon's ass all over the place, even after Eragon's elven power-up. Even the fact that he had just done it to the protagonist, did little to diminish the awesomeness.
Thorn also cemented his badassery by holding his own against and wounding Saphira even when he was severely injured and Saphira was armored.
The only part that diminishes it was that Murtagh had just entered the battle, while Eragon and Saphira had been fighting for hours and were tired.
Sloan's calling out Eragon in Brisingr is a truly magnificent What the Hell, Hero? moment, and Eragon's reaction subtracts the "Hero" part from the Sociopathic Hero normally associated with him.
Elva gets a similar moment later on in the book. When Eragon attempts to strip her powers from her, she calls him out for his self-righteousness, and points out that he has no right to do something like that to her simply because he didn't approve of her. "That way lies the depraved pleasure of controlling others for your own satisfaction. Galbatorix would approve." Eragon is so shocked that he is brought to his knees and is unable to even phrase a reply, because he can't think of anything to defend himself against her accusations.
And after Elva leaves, Angela slaps Eragon in the face. Yes, she-the eccentric human spellcaster-slaps Eragon, the Dragon Rider with the super-fast reflexes and enough strength to smash through an armoured man's chest. (Well, "human".)
This also counts as a moment for the hero himself. If he were like the elves, he'd probably just ignore Elva, but he was willing to at least listen to other people.
A soldier in Feinster continued the example when he confronted Eragon, by refusing to back down, standing his ground against Eragon despite his elven powers, and calling him a selfish glory-seeker. "Well, I curse you, Shadeslayer! I curse you with all my heart! May you leave this land and never return!"
Roran, when in Brisingr he defied poorly formed orders from his commander to win a battle and kill nearly 200 soldiers by himself.
In previewing Book IV, Christopher Paolini actually used the term "crowning moment of awesomeness" in relation to Angela, stating “There is a lot more to Angela than meets the eye. If you like her, you’re in for a treat with Book Four. Angela gets several crowning moments of awesomeness therein.”
Oromis and Glaedr not only battling Murtagh and Thorn above Gil'ead, but also holding their own before Galbatorix intervenes. Another mention goes to the elves, who enter the war against the Empire not by marching to join the Varden or sending out messengers, but by capturing one of its major military bases. And then mocking frickin' Galbatorix.
Only mentioned, but the Old Rider Thuviel, who magically converted some of his body's mass directly to energy in the Battle of Doru Areaba, horribly killing himself and poisoning his home with radiation, just to conceal the last dragon eggs and Eldunarya from Galbatorix.
For those out there who don't know what this means, this guy basically just turned himself in a frickin' nuclear bomb. A nuclear bomb.
Murtagh using the True Name of the Ancient Language to strip away Galbatorix's wards.
Eragon defeating Galbatorix, by making him feel the pain he has ever inflicted on anyone or any dragon, inadvertently or not, over the century he has lived.
Nasuada getting captured by Murtagh and withstanding massive amounts of pain and torture from Galby himself without giving in. And when Galby starts making her hallucinate, she plays along, and laughs as she goes through crazy visions and horrors.
Roran capturing Aroughs. The guy had 3 days to capture the nearly-impregnable city, only 800 warriors (with an insubordinate commander) + several truster villagers and a magician of modest abilities. He slept through the first day. He spent the 2nd day thoroughly studying anything of use for his plans. Then on the 3rd day, he and a force of warriors near single-handedly snuck into the city's waterways with barges, and captured it's center, through nothing more than sheer determination.
And before this, when a small force is charging towards their camp, he has every single person besides himself hide. Some time later, this very same force retreats away from him, a single man. He scared an entire cavalry force away with nothing more than words, sheer confidence, and a very simple magicians trick.