Auron has tremendous awesomeness, but nothing defines his character like him standing still in an arena where a group of monsters are attacking while a panicked crowd run past him, making a hmph sounds as a huge dragon-like monster roars at him, then killing it with one attack after the FMV ends. Watch the FMV scene. Keep in mind, this was really his introduction, and yet he doesn't disappoint the rest of the way.
The scene where Yuna, after a lifetime of preparation for becoming a summoner and defeating Sin, and having undertaken her pilgrimage ready to die to give the people of Spira a temporary relief from sorrow, finds outthe truth about the Final Summoning used to defeat Sin in the past and promptly says "Screw Destiny, we're going to end the Vicious Cycle permanently, not just start it over" was her Crowning Moment Of Awesome. The rest of the party quickly follow suit, but it was her moment.
Yunalesca: Now choose. Who will be your fayth? Who will be the one to renew Spira's hope?
Yuna: No one. I would have gladly died. I live for the people of Spira, and would have gladly died for them. But no more! The Final Summoning...is a false tradition that should be thrown away.
Yunalesca: No. It is our only hope. Your father sacrificed himself to give that hope to the people. So they would forget sorrow.
Yuna: Wrong. My father... My father wanted...to make Spira's sorrow go away. Not just cover it up with lies!
Yunalesca: Sorrow cannot be abolished. It is meaningless to try.
Yuna: My father...I loved him. So I...I will live with my sorrow, I will live my own life! I will defeat sorrow, in his place. I will stand my ground and be strong. I don't know when it'll be but someday, I will conquer it. And I will do it without...false hope.
And there's Auron's lines, of course.
Auron: "This is it! Now is the time to choose! Die and be free of pain! Or live and fight your sorrow! Now is the time to shape your stories! Your fate is in your hands!"
Wakka also nails it with this quote from the same scene:
"Hah! I'd never forgive myself—no way! Not if I ran away now. Even in death, ya!"
That whole scene was a gigantic Crowning Moment, really. For your viewing pleasure, check it out.
When the party interrupts the marriage between Seymour and Yuna... with an airship that's currently going down in smoke. Need I mention the chain-surfing? Or the giant guardian dragon? Doubles up with Crowning Music of Awesome.
How about Yuna jumping off the building and summoning Valefor in midair? That freaking rocked!!!
A subtle awesome moment is when, before Yuna jumps, Seymour tells her "If you fall, you'll die!", Yuna simply wipes her mouth with her hand, unfazed - as if to say his forceful kiss didn't break her.
The sheer certainty and resolve in her voice as she says this line is enough to make you actually believe it.
Lulu's reaction to when Tidus says "You're more my type, Lulu."
Lulu: Interesting... I suppose I could consider adding you to my list. I wish you good luck, little boy. You're going to need it.
YMMV, but when Seymour changes into Seymour Flux on Mount Gagazet, his speech to Kihmari and his transformation changes him from a soft-spoken wannabe to a AAA badass. Not to mention actually defeating him is a Crowning Moment for the player.
Seymour's introduction via Authority Equals Asskicking. In the middle of the stadium being swarmed by fiends, Seymour calmly walks onto the dais, makes the Yevon prayer, and summons Anima, and promptly slaughters all of the fiends in the area in less than a minute, all without harming a single human being.
Not so awesome when you find out that Seymour and the Guado were responsible for the fiends appearing in the stadium in the first place.
He gets another one when you fight him for the first time, cementing himself as a Magnificent Bastard when you choose to have Tidus talk to him:
In a game with the first voice acting venture in Final Fantasy history, the production team could have easily given Tidus a rigid name and made life easier for themselves. Not only did they let you name him, but they wrote the script in such a way that the gameplay doesn't suffer for it. They had to bend over backwards and push Hey, You! to its limits, but they did it.
Lulu's line the first time she casts Flare.
Lulu: KNOW PAIN!
Even better if you manage to get her Ultima, where she repeats her Flare line... followed by a spine chilling:
Lulu: ...but not for long.
Though it barely has much use in-game, her line as she casts Death is also awesomely chilling.
Lulu: Nice knowing you.
Another one for Lulu; "A day with you is never dull!" - followed by her suddenly diving off the side of the airship.
Nearly every character got a CMOA line if they're the ones to speak up at that point.
Kimahri Ronso, stepping up to Biran and Yenke solo, and winning. Biran being the Ronso who, apart from towering over him, is the reason he's hornless. Imagine the pride he hid after the fact. Kimahri's honor was finally restored after ten long years. the leading up lines were also badass.
Kimahri: "Then I must prove my strength!" "This time, I win, I will win."
The battle is designed to be awesome no matter what Kimahri's stats are, as Biran and Yenke's stats scale to his. So whether you're just playing the game straight or decided to go on a grinding spree or ten, the rogue Ronsos will still be happy to provide you a challenge.
Here's one for the player: Obtaining the Mark of Conquest. It's a key item you get for defeating Nemesis in the Monster Arena. A Bragging Rights Reward, to be sure, but for all that effort, you really do deserve to get a little something out of it.
Really, beating most Creations in the Monster Arena counts. You take on a hideously overpowered enemy with far more HP than both forms of the Final Boss combined and has any number of gimmicks such as Super Speed, 100%-chance immunity-proof status effects, buff-cancelling counterattacks, and six-digit health regeneration. And you kill it anyway.
Yuna gets a subtle one when Dona insults her for having so many guardians and brags that she requires only one. She calmly responds that she only has as many guardians as there are people she can trust with her life. When you later realize this is considered the common definition of a guardian in Spira (Seymour's talk about "The Code of the Guardian", for example), then Dona having only one becomes something she shouldn't be bragging about: Given her abrasive way of dealing with others, she's probably alienated so many people that she only has one person left she can trust. In other words, not only did Yuna calmly counter Dona's insult, she threw it back at Dona without her realizing.
Tidus in the later half of the game after Character Development. He finds out that he's just a dream of the Fayth and if Yu Yevon, who keeps Sin alive is destroyed, the dream will end and Tidus will effectively die. Tidus accepts this without hesitation and goes ahead with destroying Yu Yevon and right before Yu Yevon is fought, he tells the party that he'll disappear and right before the battle starts, he says "I know it's selfish, but this is my story!" and process to kick Yu Yevon's ass.
Oh, you were going to use an Aeon on Seymour Natus or Seymour Flux? Too bad. The way he taunts you before busting out his Aeon-exclusive One-Hit Kill is what sells it:
The party and the airship sending Sin crashing down in Bevelle. Everyone on the surface is in awe at the sight.
The final sequence of fights with Sin counts as one for the party. The massive creature that a thousand years has stood undefeated except for one method, and you beat it by yourself. For the first three phases (both fins and the core), rather than the usual ominous "Enemy Attack" boss theme, the game instead plays "Assault", as if saying "Sin's caused death and destruction for a thousand years and it's time to make it pay."
Even moreso when you fight Jecht, Braska's Final Aeon. If this could beat the undefeatable Sin, the fact that you kill it makes you more badass than anything else in the game.
Minor example: Seeing the crowd chant "Wakka!" in the blitzball game.
One of the player if they manage to beat the Goers and win the Championship for the Aurochs. Causing the worst blitzball team in all of Spira to take down the equivalent of the world champions is the epitome of CMOA.
Doubly so if the player pulls it off without relying on the Jecht Shot and/or Wakka ends up being the one to score the winning goal.
The final battle with Seymour wasn't awesome enough you say? How about using Anima against him? It even cues special dialogue from Seymour when it is summoned against him. Further, defeating him with the aeon created from his mother's fayth, allowing her to atone for setting him on the wrong path, summoned by the woman he's been obsessed with all game adds a layer of poetic irony/justice to this creep's defeat. Using Oblivion to strike the final blow will only make it that much sweeter.
Seymour is no slouch either when he sees what he's up against. He knows what it can do first hand and he is generally unfazed.
Seymour: You would oppose me as well? So be it!
From his first line, which is a Badass Boast, Yojimbo cements himself as a walking moment of awesome in everything he does.
Fayth: I am the blade of vengeance. They dare only whisper my name: Yojimbo.
Defeating any Bonus Boss counts for the player, as even Dark Valefor- the easiest- requires hours of grinding and preparation while Penance can take days!
The death animations as those Dark Aeons fall. You've seen them lots of times with your aeons most likely throughout your grinding efforts. But they probably never made you feel as awesome as they do once you see them and realize you did that to them. Just to pick one, the sight of Dark Bahamut falling to its knees in front of your party is not a sight soon forgotten.
Completing the Catcher Chocobo challenge. Aside from finally being able to complete Tidus's Celestial Weapon afterward, try running the race again. If you beat her in the race, she repeats her standard line, "Don't worry. Someday you'll hold the record!" but with one additional sentence: "Oh wait, you already do." There's just something satisfying about the game acknowledging all you've been through to get it.
Grinding like hell until you can have Lulu's little Onion Knight bash all that stands in its tiny little way for 99999 damage. Or even the soft metal-clinking sound Yuna's Nirvana makes as it deals the same damage. Or a sweet form of catharsis: having Auron with Masamune use Shooting Star on an ambushing Great Malboro and watching it fly off the walls of Omega Ruins.