Maximilian Jenius' first battle Super Dimension Fortress Macross/Robotech. When Kakizaki fails to live up to his boasts and has to be rescued by Hikaru, the two lose track of the more modest Max. They find him destroying battlepods left and right with deadly ease.
It should be noted that the above battle was Max's first ever combat mission, and yet he downed nearly twice the number of enemies as his commanding officer, series protagonist Hikaru Ichijyo.
Max gets a second in Macross 7, where he orders the Macross Cannon fired. At empty space. When queried on it, he replies with something along the lines of 'call it the intuition of a genius'. The cannon is fired and an enemy battlegroup folds in directly in the path of the beam. Made this lurker's jaw drop at the sheer awesome.
Also notable in Macross 7 for absolutely stealing the spotlight from Gamlin's 'death' by folding directly into the middle of a stalemated battle, outflying every single ace in his space-force (despite being in his 50s), shooting down just about all the enemy VFs, flying down a tunnel while being shot at with lasers, and bulls-eying the Big Bad in his secret lair, all within about 30 seconds and without anyone understanding who had just shot down the entire enemy force.
Let's not forget that in-universe, Max is the first person to marry a hot alien babe, who gave birth to seven (!) daughters.
The way he proposed to said wife also counts, she attacks him with a knife and Max disarms her and then asks her to marry him.
Don't forget their marriage either. Remember the scene at high-profile marriages where a line of guys cross their swords above the couple? Macross does it with Valkyries in Battroid mode crossing their guns' muzzle flashes. And don't forget about the Macross-shaped cake either.
Oh, and the jaw-dropping dogfight with Milia, both in the series and The Movie. Max came out on top in both cases, proving that by beating the best Zentradi fighter pilot (a female no less), he's technically the best pilot in all the Macross universe.
In one of the Robotech: The Sentinels novels, he and his daughter Dana fight. The fight is interrupted before it concludes, but it's implied that she may have managed to beat him if they'd fought to the finish.
Misa gets her tactical moment, coinciding with the Macross' super robot moment in Ep 06 "Daedalus Attack". After the main gun fails to fire due to the interference from the pin-point barrier system, Misa gets the pin-points to concentrate on the bow of the Daedalus (right arm) and punch through the bow of Zeril's battleship. The Daedalus opens up and all 587(!!!) destroids open fire into the innards of Zeril's ship. No other anime has this and no other should.
The Robotech novelizations turn the Destroids dying horribly in Khyron's assault on the earthbound SDF-1 into an awe-inspiring heroic last stand as the Destroid pilots try to buy time for the VFs and MAC/Monsters to arrive; they also include a scene of the Destroids marching out to back up the SDF-1's turrets in the final battle against the Zentradi fleet that was not in the anime, describing it loving detail and crediting them with tripling the SDF-1's short-range firepower for the last battle.
Hikaru's moment comes during the epic "Force of Arms" "Love Slips Away" episode, which starts with the near-destruction of Earth and ends with the destruction of the main Zentradi fleet. While his Valkyrie disintegrates, he returns to Earth to rescue Misa.
And, of course, the "Do You Remember Love?" final attack run from the namesake movie is a Crowning Moment for the whole medium of anime.
Breetai gets thrown into space in the episode where Hikaru boards his ship. Breetai manages to walk across the ship to another airlock and then beat down two Valkyries with a pipe.
A UN Spacy Zentradi infiltrator soldier makes a fake distress call to Kamjin and gets his cigarette lit up by allowing a Valkyrie fire his assault rifle in order to use the muzzle flash as light.
The atmospheric entry sequence in Macross Plus, set to Yoko Kanno's "Information High." Shortly followed by the final Isamu/Guld dogfight, the duel with the Ghost fighter, and Isamu rejecting Sharon's "Faster! Faster!" illusion when he hears Myung sing to him, the last half hour or so of Macross Plus is a giant Crowning Moment Of Awesome.
The intro is no slouch either, though for a different kind of awesome. The scene is a prologue, set several years before the series begins proper with the music set to Voices, arguably the main theme of the show. How the scene is set up is awesome in an artistic sense as it takes "show, don't tell" to its logical conclusions. See here.
Shin Kudo gets one at the beginning of the first episode of Macross Zero, where he out flies an Anti-UN Valkyrie, in a F-14. He only loses because the other pilot pulls tricks he has no idea are even possible (transforming). He then fights that pilot again in the last episode, with about one month's training in a VF-0 and once again manages to out fly his opponent.
Episode 7 of Macross Frontier at first appeared to be a crowning moment for the entire goddamned series. It must be seen to be believed. However, a short while later, Episode 14 came along and comprehensively one-upped it. Little did anyone realise, of course, that the real awesome was all being saved for the finale, twenty five minutes of the most intense space combat and wildest emotions ever set to animation, in which every single character and mecha gets a personal Crowning Moment Of Awesome.
Ranka in Macross Frontier, episode 12. She pierced the heavens fold distortion with her drill special fold engine, descended from space onboard a fighter plane sporting four huge-ass loudspeakers under its wings, landed in the middle of a battle between two Zentradi armies, and proceeded to pacify them through the power of pop music. Not bad for an Ordinary High-School Student. Memo to Celestial Being: THAT'S how you stage an intervention.
Pretty much anything Sheryl does to assert herself, beginning with her amazing recovery in episode 1 when Alto crashes into her, and she incorporates it into her act (making him utter "You're incredible" in awed tones,) all the way to singing "Diamond Crevasse" for the panicked refugees in episode 20, who were crying out for Ranka moments before.
"Ranka ... if you are the songstress of hope, then I will try singing amidst despair."
Brera defending Ranka from the Hydra. Just as it's about to jump on her, he gets in front of it and it bites his arm. He then pulls his arm away and uses his other hand to punch the thing through a wall. We find out a couple of seconds later that he's a cyborg, so the Badass factor is lessened somewhat, but it's still pretty frakking awesome.
Special note here needs to go to Canaria Bernstein, since she was the one who allowed for the previous moment of Awesome by destroying Galaxy's shield generator with her Koenig Monster.
The SING-OFF in the hospital atrium balcony between Sheryl and Ranka for Alto's attention/affections...
Personally, I found that moment simutaneously totally awesome and completely hilarious; Alto is obviously terrified of the two women fighting over him.
Ozma's burst of Hot-BloodedUNSTOPPABLE RAGE in episode 17, set to a Fire Bomber classic (which the others promptly lampshade). Further noteworthy for his surviving his injuries, which were accompanied by a slew of references to Roy Focker's death in similar circumstances in the original series. Even the writers were in on it, as the blatant scene had characters immediately commenting about how emotional everyone would be if that had happened, and Ozma looking directly at the camera when he delivers the line, "Didn't I tell you? I will never die."
Another of Ozma's: "I'M NOT AN ADULT, I AM A MAN!"
My first thought upon seeing that scene: "When the hell did Kamina get here?"
It's worth noting that they share the same seiyuu.
Brera's Establishing Character Moment in episode ten, where he calmly gets a chunk of his arm torn off by a Hydra, then proceeds to FALCON PUNCH it across the area he's standing in. When it charges at him again, he cuts it in half with an arm blade and calmly walks off.
The final battle: Holy crap there's SMS charging in guns blazing with the Battle Galaxy launching psycho Ghosts killing everything and the epic songstresses unite magic girl power with the Brera Alto epic Mecha RAAAH! charge killing everything that moves with blowing off super dimensional bugs heads and the super dual Macross DEADALUS ATTACKS! Yeah, it was awesome. Hardcore shippers on the other hand had a different reaction...
Episode 20. One of the most dramatic moments in the entire second half, where Sheryl overcomes her personal suffering to give hope to everyone trapped in the shelter. The fact that it's set alongside Michael's heroic death makes the trope apply doubly.
Klein's crowning moment is when, after her Queadluun-Rea is knocked out of action, she comes back in Michael's VF-25G, piloting it in her micronized form.
One in Episode 4 with the mock dual between Alto and Klan. Set to "My Boyfriend is a Pilot" with choreography. You'd think a heart being drawn in a heated dual with cheesy J-Pop in the background would be absolute Narm. It's not.
The time Mikhail makes that shot saving Klan Klan from Brera in episode 9 also counts, even though he doesn't shoot the head off, the fact that he was able to make a shot to a Valkryie that was close to Klan's head, makes it pretty amazing.
Episode 7's duet with Ranka and Sheryl counts for both girls, as they sing knowing there are people out there fighting for the safety of Frontier.
In the movie The False Songstress:
Ranka willingly using herself as bait for the Vajra. When she's captured, this leads to moments for Alto and Michel:
Michel, seeing that his sniping position isn't good enough, flies out to an exposed position on the bridge and shoots the Vajra while Alto is grappling with it. Unfortunately, this enables another Vajra to come up behind him, reenacting an incident from the TV series... but this time, he survives!
When the Vajra drops Ranka, Alto dives out after her in his EX-Gear. Then that gets damaged... but his Valkyrie flies down on autopilot, with him and Ranka managing to land safely in the cockpit.
Sivil got another Big Damn Heroes moment in episode 47, where she shielded Sound Force from a simultaneous attack from two Protodevlin, then drove them off with the aid of Basara's song.
Hey, what about Gigil? Getting the crap beaten out of him by two other Protodevlins, what does he do? He starts singing one of Basara's songs while trying to follow Sivil. Let's read that again: Gigil. Singing. Basara's song.
Hell, Basara is a walking, rockingCrowning Moment of Awesome. Firing a Valkyrie-sized speaker into the enemy ship's bridge and rocking out so hard Gepelnitch almost falls over from the sheer volume? Fuck yeah!