Leo taking out two cyclops with a robot crane, and then a third one by dropping an engine block on it.
And later on, he fights a goddess. Not a demigoddess, a full blown goddess. Granted, she wasn't exactly high level, but still, THE MAN FOUGHT A GODDESS ONE ON ONE.
Piper using Charm Speak to break Leo and Jason out of Medea control.
Jason defeating King Midas's son in hand to hand combat. Then bringing down a lighting bolt that caused rain to fall in the house, freeing all the people that Midas had turned to gold.
The confrontation with Enceladus contains many.
Leo taking out five Earthborn with construction equipment.
Piper hypnotizing the last earthborn, then cutting it's arms off. Also doubles as a CMOF due to the Earthborn's Major Injury Underreaction.
Jason fighting Enceladus despite the fact that Enceladus is a giant, and the very ground was working against him.
Leo hitting Enceladus in the chest with a bulldozer shovel.
Jason finally finishing off Enceladus with a lighting bolt from Zeus.
Jason Grace: Bad-Ass!
Jason:"I'm the son of Jupiter! I'm a child of Rome, consul to demigods, praetor of the First Legion. I slew the Trojan sea monster. I toppled the the black throne of Kronos, and destroyed the Titan Krios with my own hands. And now I'm going to destroy you Porphyrion, and feed you to your own wolves."
Leo:"Have you been eating red meat?"
Piper standing up to Drew and taking her place as senior counselor, through sheer force of personality and charm, proving herself a true daughter of Aphrodite.
Thalia makes a subtle implication that one of these might be going on behind the scenes, when Zeus/Jupiter's ban on contact with the mortal world is mentioned.
"Artemis plays by her own rules."
Whatever the Goddess of the Hunt is up to, we can assume it's pretty big.
The Son of Neptune
Percy in Son of Neptune is on an I.V. of Crazy Awesome with hints of Badass mixed in. Let's start with the fact that he talked nothing but smack to Mars/Ares, in front of the Roman Camp, of which he's the second most powerful god. The next time he meets Juno/Hera in a dream, he dunks her in the river, mouthed off, and showed no respect. He takes no lip from them.
When the amnesiac Percy and Nico first meet. Hazel had been convinced that Jason was the most powerful demigod she knew, until she met these two, and when they first encounter one another, she was terrified of what would happen if they started a fight. Percy was tensing up because he couldn't remember Nico and he knew he should remember him, while Nico was tensing up due to the fact that until then, Percy was M.I.A., and he suddenly showed up in one of the worst places for him to be be. Added to the fact is that he's running into his crush after so long, both of which made it seem like they were sizing each other up. Hazel did not want to be anywhere nearby if the two drew weapons. It's pretty awesome when you can make a trained soldier cower with a handshake.
"Again, Frank waited for the god to strike him (Percy) down, but Mars just grinned like they were two old buddies talking trash" That's some Worthy Opponent vibe going on there. It also counts as a Crowning Moment of Funny.
Percy proving that sometimes amnesia is incapable of eliminating how Badass you are, especially when you have monsters after you.
Anytime Percy is doing his awesome sauce son of Poseidon thing, opening a can of whoop-ass and drinking it down.
Percy's Batman Gambit against Phineas. He uses himself as blackmail against Gaea to force her to influence Phineas to choose the wrong vial of gorgon blood.
Let's not forget his initial moment of awesome in the Roman camp, where he broke through a legion of Roman demigods through sheer speed, strength, agility, and his less organized, wilder combat style. Sure, he wasn't there alone, but Frank stated that Percy was part of the problem keeping the Romans from organizing to fight. The other part of that problem was a war elephant.
"I think I like capture the flag."
Percy in the climax of Son of Neptune. He holds off an entire army of immortal ghost soldiers by creating a hurricane and sheer badassery until Frank can free Thanatos, then breaks a freaking glacier and survives the resulting fall no problem. He then rides to Camp Jupiter and manages to take down a giant with nothing but his sword and a statue's head scarcely an hour later.
Frank rendering Alcyoneus Not So Invincible After All by dragging him across the border to Canada. While smacking him with a shield, all the way. In the face.
Annabeth finding the statue of Athena and outwitting Arachne, especially considering she did so on a broken ankle.
And the fact that she was terrified to her absolute limit.
Let's make it clear on how she outwitted Arachne. She made her weave something that was just a bigger version of a Chinese finger trap and then tricked her into getting inside of it, causing her to be trapped inside. DOWNRIGHT HER AWESOMEST MOMENT EVER
And let's not forget that the underlying reasoning for this plan was that Annabeth and Arachne share the same fatal flaw, hubris, and Annabeth recognized this and played Arachne accordingly. Essentially, Annabeth turned her own greatest weakness into the lynch-pin that ultimately took her enemy down.
Annabeth gets one earlier while defeating a gang of nine ghosts solely through her powers of observation and knowledge of architecture
Give one up to Chrysaor too! The guy just walks up onto Argos II and Percy Jackson takes out his sword ready for a big battle. What does Chrysaor do? He literally knock it out of his hands in one blow! Not even Luke or Ares have been that good! Percy had one heck of an Oh, Crap look when he realized his Invincible Hero streak just went down the drain.
Can't believe nobody remembered to mention the way Percy just wiped the Romans after Annabeth threw her knife in the water in Charleston. Doubles as Crowning Moment of Funny.
All around the Romans, Charleston Harbor erupted like a Las Vegas fountain putting on a show. When the wall of seawater subsided, the three Romans were in the bay, spluttering and frantically trying to stay afloat in their armor. Percy stood on the dock, holding Annabeth’s dagger.
Percy: You dropped this.
Annabeth: I love you!
House of Hades
Frank becoming a praetor and leading a legion of undead Romans against Gaea's army. Getting the blessing of Ares/Mars was just the icing on the cake.
One for Rick Riordan: the revelation that Nico di Angelo is gay (or bi) and has/had feelings for Percy. In a series most associate with being for kids. Uncle Rick fears no Moral Guardians!
He posted here in his FAQ with what he wanted to accomplish with the reveal, and it's amazing.
Credit where it's due, the editors, publishers, and various in-between people could have edited it out. I don't know if they fought it or helped it along, but in the end they allowed Riordan to put an openly accepted LGBT character in a children's book.
Percy and Annabeth manage to go through Tartarus, and come out not just alive, but with their sanity intact. Though it probably help that they had a Titan AND a Giant on their side, who both know the way around.
Piper having to face Khione on her own after all her friends are incapacitated. She not only manages to curb-stomp Khione by reviving a mechanical dragon with some hyper-drive charmspeak, she delivers a memorable Pre Ass Kicking One Liner before she does it, too.
It gets better. Khione is gripping her arm shortly afterward, about to freeze her over... however, Piper proves that The Power of Love is more than just Charmspeak; she literally melts the ice with her willpower.
Piper: Still underestimating me, I see? You need to work on that.
Heck, why just heroes? Give it up for the bad guys too! Tartarus! DAMN!! He only show up in one scene but man does he manage to leave a mark. Let's just say that while the books are filled with Gods and superpower, Tartarus is the first being to show to be "god-like", as in he took out two titans with just a blow! The ONLY reason he didn't kill Percy and Annabeth is because he hasn't gotten use to his physical form yet... let's just say that Typhon really is a chip off the old block.
And back to Khione. She shows up on the ship, and the first thing she does, completely out of left field, is fire Leo in the clouds to (as far as his friends know) God knows where when he doesn't come back down.
Frank versus every monster in Venice at once. Frank wins.
Just Frank's character development in general. He goes from a fat kid who's been teased at camp ever since he came to a praetor of Rome with the blessing of Mars and a body figure similar to a football player's. Aside from his physical development, his confidence grows and he manages to accomplish so many tasks that he would have seen impossible in the second book. Plus, he manages to get over the fear of his stick that supports his life with the pouch Leo gives him! Downright, the BEST character development EVER seen in a Percy Jackson book.
Leo: I'm coming back for you, Calypso. I swear it on the River Styx.
The true meaning of 'foes bear arms at the Doors of Death'. A Giant and a Titan fighting with Percy and Annabeth against an entire army of monsters accompanied by the avatar of Tartarus.
Bob's development too. He struggles with the clash between his new self and his past as Iapetus, unsure of who he should be, until finally he decides who he truly is and chooses to fight Tartarus despite knowing he can't win, just to let Percy and Annabeth escape.
What does Claymore do when his gun proves ineffective and a monster is burning his only friend to death and the building their in? Despite being an ordinary person not a demigod, he punches the monster into a pile of dust with his bare hands!
The Blood of Olympus
Piper is prepared to take on all of the Giants gathered in the Parthenon by herself, armed only with a jagged metal sword and her charmspeak.
Reyna gets a string of them in the climax. She throws herself on an explosive arrow fired by Orion to protect the Athena Parthenos and a bunch of sleeping Greek demi-gods, unsure if her body will contain the blast. Doing so gains her the blessings of both Bellona and Athena, who transforms Reyna's cloak into an aegis. Reyna then uses it to strangle Orion to death.
"You will die at the hands of a girl."
Her Shut Up, Hannibal! moment against Orion, who mocks her about Jason not really being interested in her and Percy letting her down
"She forced herself to meet the giant's gaze. "I don't define myself by the boys who may or may not like me."
During the final battle, Leo manages to turn the entire framework of the damaged Argo IIinto a new metal body for Festus.
Leo, being the one to destroy Gaea, after she made him kill his mother by accident, trying to dissuade him from joining the quest to destroy her. Hoist By Their Own Petard indeed
“Potty Sludge! Dirt Face! THIS IS FOR MY MOTHER, ESPERANZA VALDEZ!”
Nico forcibly turning Bryce into a ghost and sending him to the underworld just by talking was this as much as it was Nightmare Fuel.
You also have to give Leo credit for successfully incorporating dying heroically into his plan to rescue Calypso from Ogygia.
Will Solace calling Nico out on his self-inflicted emo bullshit.