The very first Monster of the Week gets one. It's split in half by Mikoto's sword, and seems to be defeated. Then the upper body gets back up, rearranges itself, and attacks, and the lower body grows a new head and rearranges itself to keep fighting.
Haruka Suzushiro, a Badass Normal girl in a cast of super-powered girls, permanently won a spot in our hearts when she managed to headbutt Shizuru right before disappearing. The instant Haruka got back up on her feet and shouted "You!", stomping angrily toward her with life sparks flying everywhere... well, it's pretty much a given that something cool was about to happen.
That's not her only moment, though. It takes enormous brass (and, uh... muscle) for someone to stand up to a group of fully-armed soldiers and repeatedly kick one of their tanks...and walk away from it without hurting her foot. Even the students were going, "Holy crap!"
Pretty much all of Midori's first battle where she showed off her HiME powers.
Oh, let's give the main Power Trio their credit, too. The cake-hungry Mikoto chases an Orphan away from her cupboard in the Cooking Duel episode, and she and the other two girls track it to a secluded area, where they all combine their HiME powers (Mikoto first, then Natsuki, and finally Mai) to pound the crap out of it before it can land a blow. Sweet. (Of course, Mikoto never did get her cake. ;_; Although considering the judges of that Cooking Duel ended up literally hospitalized, maybe it's best she didn't get it after all.)
Did we mention the part in the first episode where Mikoto split a friggin' cruise liner in half?
Natsuki's Big Damn Heroes moment in Episode 14. Shooting the guns away from a handful of armed soldiers, blasting them with a flash cartridge and then going all River Tam on them? Priceless.
Natsuki's decision to sacrifice herself in order to defeat Shizuru by ordering Duran to destroy both himself and Kiyohime. Her speech before she tells Duran to load his silver cartridge is both unexpected and moving.
Mai's greatest moment comes at the end of the big battle with Searrs, in which she and Kagutsuchi fly straight into a Kill Sat's beam, and not only succeed in blocking it but keep going up into space to take out the satellite itself. Oh, and we can't forget that the process involved getting disintegrated, and resurrecting anyway.
One of the most powerful moments involving Mai is in episode 3, when she accepts Kagutsuchi, and ends up being frightened out of her wits at what she's just done.
Shizuru gets one of her own when, in her first battle with Nao, she bests her without breaking a sweat, and leaving half the cliffside sliding into the sea after one stroke of her naginata.
After spending a good deal of the series as The Load, Yuuichi Tate proves himself a Badass Normal in the finale, taking on the all-powerful Obsidian Lord with just his kendo skills (and this immediately after returning from the dead).
Tate's greatest CMOA happened with his own death. Because of how calmly and even almost cheerfully he took it, since it meant the two girls he cared for the most wouldn't have to fight each other anymore. Plus he clearly knew that if either Mai or Shiho lost their fight, he'd die - but he told Mai without a single bit of hesitation to defeat Shiho's CHILD for her own sake. In the end somebody else did it, but that led to as said above, dying almost happily while knowing that Mai would be okay.
Mikoto is the only HiME said to not even need a Key because her basic Element is already so powerful. And by powerful we mean that her BFS can become an even bigger BFS.
Akane summoning her Child and curb stomping the Three Scale Sisters to save Mikoto and Kazuya.
Natsuki freezing the shells her mother's Duran shot at her and Yuuichi, then shooting off her Princess earring and defeating her while she's in the middle of taunting her.
She later one-ups herself by defeating Mai of all people by freezing an even more powerful and darker form of Kagutsuchi. That's right, she defeats both Mai and Kagutsuchi in a head-on battle.