Yuki Nagato vs. Ryoko Asakura. Particularly the moment when Asakura has just shoved a bloody great glowy knifey thing through Nagato, and Nagato's only response? "Commencing nullification of data link." Check and mate.
No more than a few seconds before, Nagato took a shot of five spears of steel right into the chest while protecting Kyon. Her ENTIRE RESPONSE is to stare at Kyon with the best confused look she can pull off, and speak in that completely deadpan voice, "There is no need for you to move. I am fine." She then calmly reaches up, and PULLS ONE OF THE SPEARS OUT. Like it's something she does every day. It turns into a desk.
More entertainingly, right after the fight, she slumps to the ground. Kyon, panicked, begins to talk about getting an ambulance. Yuki comments "No need. My body did not suffer significant damage." So, you know. This means taking five spears of steel directly through very, very vital parts in the chest, then two immense glowing spear... somethings. Which basically impale her right above the breasts. Read: lungs are FUCKED. This isn't significant damage. Holy shit.
Okay, you know what? Yuki's so stoic and starts out so one-dimensional that ANYTHING SHE SAYS is a CMOA.
Yuki's actions during the "Mikuru Beam" incident in Sigh are also pretty epic: she catches laser beams. Barehanded. Realistic laser beams at that. Meaning that, yes, she is actually capable of moving faster than the speed of light.
The climax of the Melancholy saga, a.k.a. Kyon saves the dimensions by making out with Haruhi. Not only was the tension built up perfectly, but the animation and music made the scene even more awesome. A Crowning Moment of Heartwarming.
The Cave Cricket battle, which showed off Koizumi's skills in battle as well as more Nagato fight scenes. Too bad it was short, but the BGM to the fight made it all worthwhile.
In the fourth book, which was adapted as The Movie, Kyon told Yuki to tell the Data Overmind that she was under his protection, and if anything happened to her, he would destroy it, alien god or no...with four words: "I am John Smith." Also a Crowning Moment of Heartwarming.
The subtler bit just after, where he tells it to allow Yuki to act more human, and she thanks him, is also a good candidate for the moment of the book.
In fact, in one audio special, Kyon blackmail Haruhi, with the threat of calling her "Haru-Haru" if she forces Mikuru to dress in humiliatingly revealing clothes at a concert... Haruhi's reaction was priceless and must be heard to be believed. Just goes to show, whether you are an omniscient alien or an obliviously omnipotent schoolgirl, Kyon knows how to blackmail.
Kyon's future self also has a Big Damn Heroes moment when he leads Yuki, Itsuki, and Mikuru to rescue his past self. Haruhi may unintentionally summon the weirdness...Kyon can mobilize it.
Yuki gets another one, at least in the eyes of all the Playful Hackers out there, when she hacks Day of Sagittarius III's game code in the middle of the game to turn off the Computer Society's cheats. Moreover, on Kyon's instructions, she does it without any superpowers, just her native intelligence and Rapid-Fire Typing.
Itsuki Koizumi gets such a scene at the end of the novel story Snow Mountain Syndrome, when he declares to Kyon that should his Organization ever come into conflict with them, he will betray his superiors to side with the SOS Brigade. Really now, several of the scenes in the later novel stories reinforce the concept of the SOS Brigade as True Companions.
Some lines from Psycho-Asakura are simply terrifying and awesome:
"You don't wanna die? You don't want me to kill you? :("
"[...] because whether you care about it or not, I really want you to die! ^_^"
"Does it hurt? I'm sure it does. Better enjoy it while it lasts, because that'll probably be the last sensation you'll ever feel. ^_^"
In the 7th novel, Yuki answering to her past self why she didn't synchronize: " Because I don't want to"
Haruhi hearing the Computer Society President ranting about a "DUEL!" from a distance and delivering her famous Dynamic Entry. That was pretty damn awesome as well as hilarious.
When Endless Eight... well, ended. In the anime, anyway. Although the loop got pretty torturous, if they hadn't done it that many times then Kyon's solution would have been more of a Mundane Made Awesome moment. That and the epilogue also border on Crowning Moment of Heartwarming... somehow.
It borders on that for one very important reason...Yuki. As the episodes go further along, she becomes more and more visibly bored, which for her, is the equivalent of a hair-pulling fit of despair. When Kyon does his Epic Loop-Breaking Speech, her eyes actually LIGHT UP. She still doesn't say anything, but the look on her face makes it so adorably worth it.
In the "Lone Island Syndrome" Kyon gets one by utterly demolishing Koizumi's story, demonstrating beyond any doubt that just because he's the only normal person in the group does not mean he can be treated like a child.
Bonus points for his very first argument being "Haruhi wouldn't really wish for anyone to die."
Kyon's monologue about wanting to be with the SOS Brigade in the movie. The best part of it, in a movie full of awesome parts.
Also in The Movie, the moment the Brigade is (finally) brought together counts. The music track that accompanies the scene is called, fittingly,SOS Dan Futatabi, or "SOS Brigade Returns," and is the most epic part of an amazing soundtrack.
Also in The Movie: Mikuru punching Kyon in the face counts as her CMoA. He was clearly in a desperate situation, but bringing up the mole on her breast was still a terrible idea. And then Mikuru showed us that even the resident Moe-blob has her pride!