Pretty much everything Percy, Annabeth and Grover do in this book. A couple of 12-year old kids travelling across country. With no adult supervision, fighting monsters, fighting a god (in Percy's case at least) and standing up to two more. What were you doing when you were 12?
Percy killing the Minotaur by ripping out its horn and shoving it into its ribcage.
When Percy sends the Olympians Medusa's head knowing they'd be ticked off.
Percy's battle with Ares in The Lightning Thief? Percy defeating Ares, god of war, in a swordfight. Can you imagine the strength, skill and savagery it must take to do that? And Ares wasn't even in the movie.
Technically speaking he never DEFEATED Ares, all he really did was stabbed his heel. Ares was going to finished him off, but Kronos scared him off. However considering that a 12-year-old kid with only a less than a month of training was able to hold off a god is in itself Bad Ass
The children of Hephaestus are skilled with technology and make fireworks the size of Patriot missiles. Which make for massive, animated re-enactments of the Battle of Thermopylae, Hercules slaying the Nemean Lion, and George Washington crossing the Delaware. And they launch them all off on the 4th of July.
The Sea of Monsters:
Tyson and Percy taking out those cannibals in the beginning of the book.
Annabeth stabbing her knife through a Laistrygonian giant's back, reveals herself with a gust of awesomeness and then punches Sloan in the nose.
I'm sure HALF those things Tyson said to Polyphemus in the last couple of chapters in The Sea Of Monsters qualifies as well.
There's a bit in that book, passed over in a single line, where Percy cuts arrows out of the air.
The Titan's Curse:
"Mortals," she said calmly, "Are not allowed to witness my hunt." Artemis, casually making a military helicopter explode into a flock of ravens.
The Hunters' grand entrance was pretty cool on its own, whatever Thalia's opinion of them at the time.
Learning that Percy makes a habit of rescuing injured, trapped or otherwise distressed sea creatures in his spare time adds a level to his heroism.
Zoe Nightshade steals the camp van and drives south "like a crazy person" rather than risk another life unnecessarily. (Also probably to avoid any further delays while choosing the fifth member of the quest group, which is a different kind of awesome/heartwarming.)
"You're not going to hurt Zoe. I won't let you."
Keeping in mind that Zoe has not really been all that nice to him at all, Percy's willingness to stand up to the Titan General to protect her more than qualify. Especially as he comes close to being killed doing it.
Percy voluntarily stepping in to hold up the sky in Artemis's place.
Artemis whipping out two absurdly long knives (and keep in mind that she has the body of a twelve-year-old) and ripping Atlas a new one long enough to trick him into taking the weight of the sky back.
Speaking of Zoe, taking a quest perfectly aware of the fact that you're going to die horribly (she knows how much her father hates her, and she must have known that if she was going to die by his hand, it wouldn't be pleasant) because your goddess needs you is pretty awesome/heartwarming on its own.
Apollo blatantly flaunts the "no direct intervention" thing. Screw that—Artemis is in trouble, and damned if he's not going to help her.
"No one messes with my baby sister."
Percy giving up the Nemean Lion's fur. More specifically, the way he looks at Zoe and says clearly, "I'm not Hercules." His newfound appreciation for what true heroism means is a turning point for his character. Plus, he was giving up a freaking bulletproof vest with probably a bunch of other really cool features, just because he thought it was the right thing to do. That's epic heroism or nothing is.
Bianca sacrificing herself. She's only a kid, she's barely been in this world three days. That took an insane amount of courage.
Percy convincing the gods not to kill the Ophiotaurus.
Zoe fighting Ladon single-handedly, and then being immortalized in the stars.
Nico opening a rift in the earth.
Battle of the Labyrinth
Zoe Nightshade is pretty awesome, but then we learn that the nymphs in general still harbor a powerful grudge over Hercules rerouting a river to wash away centuries of horse...droppings. And then we remember Zoe's throwaway line from TC about water nymphs. "They've never forgiven me." Zoe had been helping Hercules all along. All those labors where he actually had to think, and not just pummel things? That was her, and he never even thanked her.
Nico's final confrontation with Minos in Daedalus's workshop?
King Minos: You have no power over me. I am the lord of spirits! The ghost king!
Briares turning one hundred feet tall and killing Kampe by hurling boulders at her. Made even more awesome by the fact that he had been living in fear of her for most of his life.
The Last Olympian:
Beckendorf sacrificing himself to destroy the Princess Andromeda.
How has Percy single-handedly defeating Hades's army of Roman soldiers and redcoats, not to mention getting close enough to threaten Hades himself, after bathing in the River Styx and achieving invulnerability not gotten on here yet? Coming close was the moment in the battle for Olympus where Percy (again, single-handedly) turned the tide against a battle between the Apollo cabin and an army of monsters by cutting through most of the army himself.
Percy then taking on Kronos, the Titan Lord himself. And not only survives, which is a CMOA in itself, but makes Kronos break a sweat.
Special mention should also go to Percy taking on another Titan, Hyperion by himself and pretty much stalemating with him for the whole fight. Keep in mind Hyperion is specifically mentioned as being second only to Atlas and Kronos in terms of power among the Titans.
Grover and the Satyrs also gets a moment here for beating Hyperion by more-or-less sealing him in a tree.
Note that Hyperion is the Titan of the Sun.
The Party Ponies beating back armies of monsters with NERF baseball bats, arrows and paintball guns laced with celestial bronze. Whilst screaming insults at each other over who has the better chapter.
Annabeth landing the Dare Enterprises helicopter. Without actually knowing how to pilot it.
Silena convincing the Ares campers that she was Clarisse.
Clarisse's Roaring Rampage of Revenge after Silena dies, which consists of killing a freakin' drakon, and then holding back the entire army while dragging its skin behind her.
Clarisse:I AM CLARISSE, DRAKON-SLAYER! I WILL KILL YOU ALL!!
What, nothing for Nico? He was one badass twelve year old when he showed up with his skeletal army, made the army of Kronos part for him, and pretty much saved the day.
Percy's stepdad, who can't even see through the Mist (the stuff that makes monsters and so on Invisible to Normals) grabs a sword and duels one of the monsters, and his mother blows one away with a shotgun... but what makes this truly awesome is that when Percy asks when she learned to fire a shotgun, she calmly responds, "About two seconds ago."
It's a very brief moment, but after Percy knocks Luke's sword into the hearth fire, Kronos goes to fetch it, only to be horribly burned by it having become red-hot in the coals—and then an image of Hestia flickers in the fire, frowning at him in disapproval. Yes, she stood up to, wounded, and contemptuously dismissedthe wicked father of the gods himself. So much for being "the least of the gods", eh?
Percy and Nico, for convincing their immortal god dads to set aside their personal agendas and pull their Big Damn Heroes moments against Typhon and the Titans.
Percy exacting an oath from the gods to never let any half-blood children go unclaimed again, and allowing all the gods (and their children) to have seats on Olympus and cabins at the camp.