The very first story had the team unite as one to take down Starro the Conqueror, a giant alien mind-controlling starfish.
The JLA Classified miniseries New Maps of Hell has the heroes face off a millions-of-years old self-proclaimed deity in a whole series of scenes of awesome, but once they discover its true nature the Martian Manhunter disdainfully informs their enemy what they think of it.
"We're the Justice League. We've beaten up real gods and made them cry. You are nothing to us."
He gets another great line during Morrison's run, when the heavenly host invades. The guy can talk down to freaking angels:
"You people. How many times must we face your kind? Would-be conquerors and master races, so full of your own superiority, you will not see reason or hear the voice of sanity. But see this, hear this....We will not do as we are instructed."
In the same storyline, we get to see what the Speed Force and a Green Lantern ring can do together. That's why we have the Justice League.
J'onn's refusal to adress Lex Luthor as "Mr. President" was a small one, but still cool. Also, Kyle Rayner called the President of the United States a "butt-hat". You can't tell me that's not awesome.
JLA gave Batman his moment of awesome when the rest of the heroes, and pretty much the entire world, are captured by the invading White Martians. With the message "I know your secret!" left behind in flaming letters written on the unconscious body of one of the Hyperclan, Batman turns the supremely confident invaders into confused, paranoid wrecks. What makes this scene so spectacularly awesome is that, at this point, the entire JLA has been defeated and captured. The leader of the Hyperclan suspects that Batman had died when the Batwing he was flying in crashed, and isn't worried by the thought of Batman surviving. "Batman? Batman! He's only a man!" A "man" who has figured out that they're White Martians and starts easily dispatching them with fire traps.
The smirk on Superman's face when he heard Protex go into his "He's Just One Man" rant was hilarious. You could just imagine the thought balloon above Clark's head going "I'm going to enjoy watching Bruce screw with you far, far more than I really should."
In one Justice League comic (Justice League of America, vol. 2 #14 or thereabouts), the entire JL is captured by Luthor and his Legion of Doom, and he's about to execute Supes, when Firestorm transmutes the Kryptonite knife into a butter knife. The entire Legion turns on him, until he reveals his "Trump Card": He slipped Batman a ball-point pen. By the time the Legion turns around, Bats has freed the entire rest of the League.
Batman: With a ball-point pen, I can pick the locks at Fort Knox.
In the Justice League limited series DC: The New Frontier, John Henry's entire presence in the story is not only gut-wrenchingly tragic, but Bad Ass as all hell...until the point where it turns gut-wrenchingly tragic. Steel was never the most fascinating character, but knowing what really inspired him (let's just say it wasn't Superman) almost makes this the Crowning Moment of Awesome for both characters.
Seriously, how was there not a John Henry mini-series after New Frontier came out?
Wonder Woman in New Frontier Classified. After a man douses her with alcohol and sets her on fire, Wondy responds by ripping off her flaming breastplate and smacking him with it. The only sufficient response to this is probably "Holy shit!"
...and the story even had a great historical sting by virtue of Diana doing that stunt at the very Playboy Club that had the famous reporter/feminist activist Gloria Steinem, undercover as a Bunny, there to record it all.
Grant Morrison run on JLA is made up of back-to-back Crowning Moments of Awesome. Batman is able to expose the Hyperclan as White Martians, and is the first superhero to make them truly scared. The Flash manages to catch up to a rival speedster and punch him into space in less time than it takes a vase to hit the ground. Superman wrestles an archangel immediately after restoring the moon to its orbit. In a dark alternate future where Darkseid's taken over, Batman spent seven years under torture at the hands of Desaad and won. In the same alternate timeline, The Atom kills Darkseid. Connor Hawke defeats a man bordering on nigh-omnipotence with one of his dad's old boxing glove arrows. And, in the end, the entire population of Earth is given superpowers and willingly joins the JLA and an army of angels in fighting a potentially galaxy-killing menace. You'd think that last one would be a huge event that would've shaken things up in The DCU... not so...
Grant Morrison's final issue of JLA, that saw Wonder Woman leading the entire population of Earth into space to do battle with Mageddon, was one of the first comics this troper ever read. I had no idea what was going on, but I knew it was awesome. And to this day, I can't read that comic without feeling a chill at the sheer magnitude of the story. Author's crowning moment of awesome?
Oracle gets a great moment during the mobilisation of the Earth as well. She's the one who delivers the rallying cry to everyone when the JLA gives them all superpowers, and on the last line she stands up from her wheelchair.
Batman's second fight with Prometheus. Prometheus's helmet lets him upload skills. The first time he fought Batman, it was with the skills of the twelve greatest martial artists on the planet (including Batman himself). The second time, Batman switched the discs, giving Prometheus the physical abilities of Professor Stephen Hawking. "First time I've ever hit a man with motor neuron disease."
Which leads to this exchange with Huntress: "Did I just see you cheating?" "Winning."
Zum (taunting): "You must be Aquaman. What can you do? You can't fly or run fast, can you? Your skin may be tough, but not so tough I can't just... cut through. What can you do, apart from talk to fish?"
Aquaman: "Let me think.... I can locate your brain's basal ganglia, the part inherited from your marine ancestors... and, just for starters, I can give you a seizure."
The Squire basically forcing Batman to work for her in JLA Classified.
When the entire Justice League was killed on a time-travel mission, a prerecorded video from Batman assembled a new team. During the speech he recorded some unknown time ago, Green Arrow starts complaining to the Atom - and the video tells him to shut up. That's how Crazy-Prepared Batman is.
Leading this team is Nightwing, who manages to save Tokyo and lead the group while suffering from the crippling fear that he can't measure up to Batman. Despite this, he faces down the Big Bad (and I mean the BIG Bad) with no trouble. And when she tries to kill him, the replacement League shows her... Nightwing was using himself as the bait.
Nightwing's speech when he takes his (surrogate / adoptive) father's place in the JLA would also certainly count.
Nightwing: Not Here. Never here. You got a grudge? The guy across from you stole your wife? Just having a bad day and want to take it out on someone? There's fifty levels of self repairing Watchtower you can trash to your heart's content. But you leave it outside this room. This is where the Justice League sits down to work. The. Justice. League.
Which prompts the following line from Green Arrow(to himself):
Green Arrow: When did he get "channeling the dead" vision? He's got the Bat's voice and everything.
Yet another one from the Hyperclan arc: Batman yet again showing his Crazy-Prepared, right after Wonder Woman has taken down Primaid of the Hyperclan by dragging her up into the vacuum of space.
Green Lantern: So how long can you hold your breath?
Wonder Woman: Obviously longer than Primaid. What a strange question. Why should anyone know how long they can hold their breath?
Batman: Three minutes fifteen seconds. You'd be surprised why.
It's important to note that Batman is dragging past four unconscious Hyperclan members, as compared to Wonder Woman's one, when he says this.
Because he's Batman.
Green Lantern (Kyle): Only four, Batman? You're slowing down.
In the Trial By Fire story by Joe Kelly, the JLA is battling an enemy who has telepathically manipulated the nations of the world into all-out war. After taking down a number of nuclear missiles, the JLA return to confront the villain, when he tells them that there's one more missile. The second-last issue ends with Superman racing after the missile (which was aimed for a Korean city) desperately straining to catch it... only to fail, and the entire city goes up in a nuclear fireball. The death-toll would be in the millions. We begin the final issue with, "At 13.57 local time, the town of Chongjin, North Korea, was destroyed by a nuclear warhead. The city has survived many wars, but it will not survive this. But... at 13.57 and 0.00001 microseconds, the entire population of the city found themselves standing on a hill 30 miles away. They were carried there, one at a time, by one man. The Flash. The fastest man alive." There are no words for that much awesome, there just aren't.
Daaamn would be a good start.
The time that Batman laid out Guy Gardner with a single punch.
Blue Beetle: (laughing) "One punch!"
Back to Morrison's run, and the story which introduces Zauriel to the DCU (by having him chased down to Earth by renegade angels). Leaving aside the two epic moments of (a) Superman wrestling an angel, mentioned above, and (b) Wonder Woman managing to stop a city-sized spacecraft from crashing into San Francisco and destroying the city, the most awesome moment is the part where Superman stops the moon from falling out of it's orbit. At this point, Superman had his electro-magnetic powers, and he uses those to cause the Earth and Moon to repel each other, saving both. Flash later questions him, wondering how he managed to pull off such an amazing feat. "He smiles, and it's that one smile he has, the one that reminds you he's not really from here. 'There were larger forces at work today, Wally', he says." The next panel is a wide-shot of the lunar surface, showing a giant finger-print on the surface of the Moon.
A more subtle, but still undeniably awesome moment for Martian Manhunter in this arc. First he stands up to an angel long enough for Superman to come wrestle it, but the real moment comes after. Said angel spacecraft is going to fall on San Francisco until the Martian Manhunter comes to help lift it. Impressive, but he is as strong as Superman. What makes it a true CMOA is that anyone who touches the ship bursts into flames. So Martian Manhunter is lifting a city sized spaceship while surrounded by his greatest weakness, and still refusing to give up. Bad Ass.
It comes at the very start of Final Crisis, but this line tells you everything you need to know about the League:
Superman: "These are celestials capable of cracking the planet in half and enslaving billions. Justice League condition amber."
Yet another Grant Morrison example: in "Elseworlds," the Connor Hawke Green Arrow, who has just been invited to join the JLA, shows up at the Watchtower to find that the Key has taken it over and placed the entire League in drug-induced hallucinogenic comas to power a machine that will somehow turn him into a Reality Warper. The entire issue is one, long CMoA for Connor, as he singlehandedly defeats most of the Key's robot Mooks and stops the Key from completing his plan at literally the last second, armed only with a set of his father's ludicrous Trick Arrows, which he isn't even trained in using.
New 52 JLA #1 - Green Arrow spending almost an entire week surviving in the woods, being hunted down by evil robotic versions of the Justice League, all with a serious wound to the side.
New 53 JLA #7 - Superman is captured by Rao, in a device designed to keep Kryptonians restrained on Rao's ship. Superman, after giving a The Reason You Suck speech to him, promptly starts exerting his strength. Rao tells him he can't escape. Supes replies? "I'm not trying to. I'm flying." Using his incredible strength and simple use of his ability to fly, he crashes the whole ship into the moon. As seen here.
* The climax of "Justice League Rebirth #1" with Superman pulling a Big Damn Heroes by locating the weakness of The Reaper,an Eldritch Abomination preparing earth for his race, and allowing the whole league to defeat it together, followed by the League giving it one hell of a Badass Boast that causes the creature to flee.
Batman: This is mercy. *Leave*. Tell all the other reapers, tell everyone and everything like you that this world is protected.