One scene in Macalania Temple has Yuna and company confronting Seymour. The group moves forward to guard Yuna, as it is their duty as guardians to protect the summoner (in this case, her). Yuna, however, has second thoughts about just sitting there as she gets protected:
Yuna:Maester Seymour. I trust my guardians with my life. But they are also my friends. I will not stand by and watch them be hurt. I will fight you, too!
Everything Yuna does during that scene, from going along with the wedding just for a chance to send Seymour to jumping out of the window, summoning an aeon to catch her as she falls. Yuna is made of awesome.
Yuna: Don't worry. I can fly.
Quite possibly the most emotional scene in any game, especially if you haven't translated Al Bhed (and thus don't get the conversation between Cid and the airship bridge crew) is the flight from home. You can tell from the look on Rikku's face that something is about to happen, but when the ship's entire crew begins to sing the Hymn, it really hits you in the gut. Something great and terrible is about to happen. Then Rikku tells you the reason for the solemn moment.
How about Yuna, Rikku, and Paine joining up with Leblanc and company, plus Nooj and Gippal to brave the Farplane, wreck Vegnagun, save Baralai and beat the hell out of Shuyin?
What? This far and no mention of Yuna snapping Rikku and Paine out from Shuyin's mind control at that Crimson Cave?
From a gameplay perspective, beating the Arena monsters. The harder and more controller-chuckingly frustrating, the sweeter the eventual victory.
Despite the epic storylines in the game, Final Fantasy XI somewhat spoils the moment by you remembering that most of the missions you complete have really been done with other people. Then Wings of the Goddess comes out, and we're suddenly doing all sorts of kick-ass things from pulling a Big Damn Heroes moment for a not-exactly street-smart genius kid Tarutaru, to stopping one half of an invading army force attacking Windurst. And this can all be done by yourself in the literal sense, without other PCs.
The finale of the mini-expansion, "A Shantotto Ascension," which not only reinforces the idea that Shantotto is quite possibly the most powerful being in the Final Fantasy XI universe, but it's also just... well, see for yourself.
And Ovelia's moment came when she stabbed the son of a bitch for manipulating her and Ramza.
Ramza's conversation with Rafa, Where she reminds him he isn't going to get a reward, or even be remembered as a hero and everyone will still hate him, But Ramza keeps going through everything, Just because he knows it's the right thing to do, And no one else will do it.
Mustadio gets one at the end of his introductory adventure. His skillset centers around carefully aimed gun shots to disable enemy limbs. When captured by Baert's thugs and disarmed, does he sit out the ensuing battle? Heck, no! He charges into melee and starts breaking arms and legs with his bare hands.
For my money, it wasn't. He always dies before he gets the chance to do anything, or he just punches someone.
Agrias gets a few. After leaving her and Ovelia while you go to solve Mustadio's problem, she and the princess get kidnapped. Agrias proceeds to break out of prison, and then escape from all of the enemy forces by herself. Even when you run into her, she proceeds to charge back into the enemy as you catch up to her and only quick action will allow you to share in the kills. Say what you will about the numerous Swordmagic users who join your team later, the fact she's thoroughly useful for the entirety of her tenure as a Guest character is awesome by itself.
She also gets a second Crowning Moment of Awesome (which doubles as a Crowning Moment of Heartwarming) if you take her to the Golgorand Execution Site battle. When Gafgarion ambushes you and springs the fact that the villains are Ramza's older brothers, she doesn't bat an eyelash and expresses her unwavering belief in Ramza's commitment to their goals.
When Marche is unwittingly teleported into Exodus' shrine, he plans on destroying the fruit in order to break the next seal. Babus steps up to stop him, and Cid reminds him that he's alone this time. Marche then delivers a "World of Cardboard" Speech admitting that he never had confidence in himself, and that while he can do many things in Ivalice, it isn't real and he must go home to face reality. He then proceeds to destroy the crystal alone, which nicely shows how much he's grown since coming to Ivalice.
When Ezel is cornered by Judgemaster Cid, who uses an advanced law to prevent him from doing anything, he responds by finding an antilaw to counter it, enabling himself and Marche to escape.
The best part is how cool and calm he is, even when faced with the Judgemaster himself. It's like "Oh, you're using an *advanced* law card, are you? Too bad, I'ma gonna use an anti-law, bye!"
In one particular quest in Final Fantasy Tactics A2, you have to shut down a swindler who sold bogus hammers for 20k gil, and Ezel (who you should know from Tactics Advance) shows up to give you the card that was meant for your mark - after bragging that it's worth 80k gil. So what does the card do? It's God Mode for your Judge - as Illua has the misfortune of finding out before the penultimate showdown at Delgantua, when her spell to banish your Judge is beaten off by Ezel's card like money was owed. Best 80k gil I never spent. Thanks, Ezel!
The chain of sidequests with Bowen, who seeks to destroy a giant Cocaktrice that murdered his wife many years ago. You fight the bird several times with Bowen (hume Fighter), his Illusionist, and his Raptor. Each time you seemingly KO the bird, it flees. When you fight it for the last time, you arrive to find Bowen trying to fight it alone in a blind rage. When the battle begins, Bowen's allies arrive on the scene and a different battle music plays, which is the same one used in boss battles from Final Fantasy XII. The music is only the icing on the cake. When the boss' HP drops down to a certain amount, it tries to flee again, but your clan uses an item to keep it rooted to the ground so you can finish it off for good. After the fight is over, a touching scene plays out afterward and Bowen can finally rest, knowing his quest for revenge is over.
The moment when Rufus Shinra gets up from his wheelchair, shows Kadaj the box with Jenova's head, taunts him for being a failure as a son, throws the box down a skyscraper, dodges Kadaj's enraged attack by throwing himself off the skyscraper all the while shooting at Kadaj and ending the whole thing with planting a bullet in Jenova's head definitely is his Crowning Moment Of Awesome. Adding to all of this, he pulls it off while slowly dying from a crippling plague. And keep in mind that he is a normal human without the flashy superpowers of the main cast.
Also, proving that victory is not necessary for a Crowning Moment Of Awesome, Reno and Rude get their moment in the spotlight fighting Yazoo and Loz. Sure, they get their asses kicked, but they get their asses kicked in a way that makes them look Badass for even managing to put up a fight. And they still manage to top going toe-to-toe with near invincible superhumans and surviving when they realize that they don't stand a chance in a real fight and decide to up their ante. Cue Stuff Blowing Up.
The climax of the fight against Bahamut TREMOR. The entire party performs a Fastball Special using Cloud as the projectile (yes, all eight of them, including her), tossing him straight up a skyscraper towards Bahamut. Cloud rockets into the sky, flies right through a giant blue nuke-ball breathed at him by Bahamut, lands on his back, plants his sword right into the back of its neck, and berforms a Sword Drag across his entire length, ripping a bloody line right through the dragon.
This is only in the Complete version of the movie. After fighting Sephiroth for hours to no avail, Cloud is exhausted to the point of collapse. He manages to get a bit of a breather on top of the ruined Shinra building. And then Sephiroth shows up, beats him up a little more and kicks him into a still standing wall, crumbling it and sending up a large cloud of dust. He gets ready to charge in there and skewer Cloud one last time, when the dust fades and shows Cloud standing there, defiantly holding his sword, Battle Aura flaming and ready to kick some ass. Of course, Sephiroth promptly skewers him anyway, but it was still awesome.
That's not just "kicking some ass", Cloud uses the (regular) Omnislash against Sephiroth, who blocks every blow. And then, with Cloud skewered, he remarks "Is this the pain you felt before, Cloud? Let me remind you. This time you won't forget." and grows the titular single wing. Now I'm a man, but when that last bit happened, I swear I screamed like a fangirl.
In the final fight, Cloud reveals the ace up his sleeve in the fight with Sephiroth and uses his new Omnislash. Sephiroth can't even get one full sentence out before vanishing. Turns out that Cloud has improved since their last encounter, whereas Sephiroth, being dormant, was standing still. Looks like someone Took a Level in Badass.
When the Final Fantasy main theme was played at Tour de Japon, the Crowning Moment and Crowning Music tropes became one. Halfway through the song, Nobuo Uematsu steps up to accept the audience's applauds. The conductor offers Uematsu the conductor's baton and he at first seems to polietly refuse, only to reveal he has his own with him and step up to lead for the rest of the song.
At about the end of the third hour into Final Fantasy Crystal Chronicles: The Crystal Bearers, "Goldenrod", the Yuke that Layle's been pursuing since the start of the game, decides that she's given Layle enough warnings to keep his nose out of her business and tells him that she'll demonstrate her true power: Summoning. She promptly creates a summoning gate that releases, of all Summons, the dragon god-king himself, Bahamut. Not one to be outdone, Layle has a boss fight with the dragon who is traditionally the most powerful summon ever. However, when telekinetically launching rusty swords through his armor does little more than anger Bahamut into destroying the arena, he needs to find a new tactic. So what does Layle do to harm Bahamut? He performs a horizontal Colossus Climb on Bahamut while he's in mid-flight, grabs Bahamut's tail (which is a gigantic spear in this incarnation), and skewers Bahamut through the heart WITH HIS OWN TAIL!
Just a general one, but Limit Breaks and Summons. You know they are awesome. You see an Omnislash or Ifrit coming your way, you know that you are pwned.
Agni's Philosophy showed up at E3 2012 with absolutely no prior teasing or fanfare, and proceeded to show everyone exactly what Square Enix have been beavering away at in preparation for future consoles. Keep in mind that this is all in real-time.