In MÄR, there are quite a few of these, especially during the War Games:
Ginta's first fight against Girom, in the underground lake. After fighting an even battle against Girom and not getting very far, he reveals Babbo's new Version 3: a giant flying stone gargoyle, with far more power then any guardian thus shown. After a few exchanges Girom is reduced to bargaining for mercy from Ginta, who responds: "And when the people of Vestry asked for mercy... what did you do? Here's my answer!!!" and he then has Gargoyle plow Girom into the ground. Then he proceeds to summon even more of Gargoyle's power, sending out a huge beam of power that plows through feet of solid rock wall, big enough for a huge ship to sail through. All of this ends up thoroughly impressing the Big Bad. And this is before the War Games.
And the fact that he sends Girom flying again the next time they fight in the 5th round using the same reason 'Fly away and repent' just makes it more awesome.
While Nanashi gets the Idiot Ballfrequently throughout the series, he still gets a few CMOAs; one of the coolest was against the Chess Piece Loco. She has him paralyzed with a curse, and then begins to harm him with a huge voodoo doll. 2 or 3 spikes is supposed to be enough to kill a man, but Nanashi survives all five. He breaks free of the paralyzing Ärm through sheer force of will. And then he fries the straw-doll Ärm with his lightning attack, deliberately missing Loco by inches before passing out. Loco immediately understands that even though she "won" by the rules, she really lost.
Before this, around the time of Ginta's fight with Girom, Nanashi faces off against a dumb giant named Orco. He's got enormous strength and a body-hardening ÄRM which makes him as tough as stone. When he throws a huge punch at Nanashi, Nanashi stops it with one hand, without using a single ÄRM. He then punches the boulder-hard Orco into the lake, and proceeds to fry him with his electric ÄRM.
I remember an incident in Flame of Recca where a character named Joker performed a very similar move against someone. J-Keeper attacks Joker and Joker stops that huge thing with one finger as well. (He had a gravity-altering weapon with him though.) Still, there is speculation that Joker and Nanashi are the same. If it is, more crowning moments of awesome!
Then there's Dorothy. During her fight with Maira, she nonchalantly fishes for an Ärm in her extra dimensional zipper. Then as Maira sics his amoeba guardian on Dorothy, with the intent on having it eat her, she grins evily and summons her own Guardian "Toto". As the aomeba leaps at Dorothy, cracks appear in the sky, and an blur of motion grabs Vaqua and pulls it into the hole. A few agonizing seconds of squishy crunching later, a giant lethal-looking dog steps out, and announces that he ate the guardian. Then, she interrupts Maira's surrender by having Toto eat him.
In a later battle, she defeats the Chess bishop Avrute with one slice from her Ring Dagger. The knight Rapunzel was not happy with the bishop's defeat, as she effortlessly chops his head off as punishment.
And because of that one fight, Dorothy fought a second time in that particular battle, against the knight Rapunzel. Despite getting impaled in the stomach, she not only survives, but defeats the ugly bitch with a single slice from her razer-sharp wind generating broom (of course, with the help of an ÄRM that generated a super-loud screech).
His first war game match against the Chess rook Pano could possibly count. After paralying Pano, Jack goes up to her and gropes her. While being watched by hundreds of spectators. Pano prompty hits him in the groin and defeats him.
What makes it so great is that he doesn't actually grope her (it looks like he intends to keep climbing for some reason), but the crowd catches on to her calling him a pervert and so he backs away. And she was just as surprised as everyone else when the Ball ends up hitting him.
The fact that somewhere between their matches, they seem to have fallen for each other just makes it even better.
Then Snow asking if getting hit there really hurts that much makes this a Crowning Moment of Funny and shows how truly innocent she is.
He fights Pano in a rematch later on, and, when Pano used her Ball Hammer ÄRM, which is basically a giant flying metal ball that can travel at breakneck speeds, Jack catches it in midair. Sure, catching it was painful as hell for Jack, but it was awesome. Jack promptly defeats Pano by drugging her with mushrooms, causing her to hallucinate and forfeit the match.
Before Jack's fight with the Chess bishop Kollekio, the latter tossed him a card. In the manga, it was originally a Joker card, symbolizing Jack's imminent death. But in the anime, he accidentally tossed him the ace of hearts. The disgusted, confused look on Jack's face was priceless, causing Kollekio to have an Oh Crap moment and search for the Joker card that he intended to toss him in the beginning. Definitely one of the funniest moments in the series, if not the funniest.
And Jack defeats him by throwing earth beans into his mouth. The beans grow into thick, spiny vines, and come slithering out fast. Let me rephrase that: Kollekio puked out an entire fucking beanstalk. It looks just as painful as is sounds.
How about Jack's fight with Knight Weasel. In addition to being another plant user, as well part of Phantom's hand-picked squad of the best Knights in Chess, he's also the one who killed Jack's father in the previous War Game. It quickly becomes clear that Weasel is the worst opponent Jack could have fought, because as a plant user, Weasel has many number of ÄR Ms capable of rotting and destroying Jack's plants before they can touch him, while keeping his own intact. So what does Jack do? He pulls out his second Caldian ÄRM, a fire guardian, just to play completely against type and surprise Weasel. Not only does the thing utterly incinerate Weasel's Yggdrasil, but Jack proves smart enough to sic the thing on the tree growing on Weasel's head. Turns out that it's the source of Weasel's power. Victory for Jack.
Alan, of course, has quite too many to count, but this troper believes that one of the best was when an overconfident Chess Piece, Alibaba, pulls out a huge guardian via a Genie's Lamp and, after a long bout of gloating, sics it on Alan, who then blows the guardian into pieces without even using an ÄRM. He then drags the rook behind him as he lectures him on the proper use of a guardian, and then dumps him into a volcano.
How the hell did this page go on without Alviss? Aside from subverting a great deal of negative tropes associated with The Lancer, Alviss is also one of the most Badass characters in the show. Especially prominent when he's fighting the Chess Pieces:
Defeating a Chess rook with a single attack from a GIANT TOTEM POLE? Check. Standing his ground and surviving against a knight, despite being completely out of his league? Check. Defeating a Cyrano/Captain Hook crossover WITH HIS BARE HANDS? Check. Defeating a Pied Piper Expy capable of sucking away magical power? Check. Defeating a Nigh InvulnerableHighly-Visible Ninja by ENCASING HIM INSIDE A GLASS BALL THAT IMPLODES? Check. Let's face it, every single one of Alviss's battles is a CMOA for him.
His flashback from the Chess War six years ago was, chronologically, his first crowning moment. When he came across the Big Bad Phantom, who had just killed several Cross Guards, he charged the bastard in sheer anger. And this was when he was ten years old.
And because of that, Phantom cursed him with a Zombie Tattoo. This prompts Alviss into training harder in order to destroy Phantom, which is the only way to remove the curse.
Although most of hers are in the anime fillers, due to her being not present for most of the major War Game battles, Snow gets some time to shine as well.
While training with Dorothy and the 'Sailor Planets' show up, she smashes herself so as not to be used as a hostage against Dorothy. She ends up proving that she's just fine due to quick thinking and helps Dorothy defeat the three Sailors they've been pitted against. And just for the record, the other pairs within the Shurren no Mon, are guys and they just barely pull out a win, the girls have three opponents, and they win rather easily while the guys struggle for their victories.
When Alvis has been turned against the team, Snow sneaks up and snipes the ARM that's causing the problem.
Before the second round, Ginta tells her she doesn't have to fight, since she's a princess and all, she responds by slapping him, and saying that she does have to fight.
When she does fight in the second round, she surprises Dorothy and Nanashi with her being smart to use the opponent's strength to her own advantage and winning the battle.
When she fights in the third round, despite being in a poor environment for her, volcano field while she manipulates ice, she fights with all her strength and refuses to give up when she's left helpless. So Alan and Ginta have to convince her opponent to accept the defeat she won't admit to.
Leading up to Alvis' crowning moment in the next field when he beats the same opponent for her, without even using an ARM.