King Kong (1933) is such an awesome spectacle that even today it stand up as one of the best films ever made and a Moment of Awesome not just for those involved in its production; nor just for Willis O'Brian creating Trope Codifier special effects and emotive performances that rival modern Serkis Folk in terms of character and performance; Nor for nearly single handedly saving the production company RKO Pictures from Bankruptcy during the height of The Great Depression. Between all it did, King Kong is a Moment of Awesome for the ENTIRE FILM INDUSTRY.
The 1933 version: the "shaking the log" scene. Take a good look at all the different elements of that scene, then remind yourself that only a decade before that scene was made, they hadn't even invented sound. And that's just one example where that occurred. The "Lair" scene before Kong battles the Plesiosaur has, using 3 different methods of compositing, one shot with mat paintings, 2 animated figures with miniature sets and live-acted figures (5 separately shot and projected, 2 actors, 2 set pieces, 1 stylistic element) all at different variable levels. Modern effects artists are in awe of that shot to this day.
Keep in mind, it took up to 15 hours to get a minute of film for simple sequence (The T-Rex fight in the original took 7 weeks to film). Worse, every shot had to be done as though they only had one take on it. The fact that they got this 18 inch puppet to truly act is a miracle unto itself! Everyone who knew the head animator, Willis O'Brian, can see elements of him in Kong's performance.
One moment in particular would have to Kong's fight with the T-Rex.
The 2005 version cranks it Up to Eleven by having Kong battle three V-Rexs at once.
Kong's rampages in all the films are definitely one of the highlights.
Weta gets a Real Life MoA for producing, purely out of love for the King Kong mythos, a re-creation of what the 1933 film's planned "bug pit" scene could have been like, had it been achievable with the technology, time, and budget available to RKO. Check it out, it's a DVD extra with the 2005 movie.
Ethnic Scrappy Charlie the Cook gets one in Son of Kong when trapped with his allies in a crevas by a territorial Styracosaurus, he breaks his meat cleaver off in the monster's FACE—causing it more anger/pain than his allies' guns had.
Arguably, a scene where Jack Black has fallen in the insect trench and all of his crew is dying around him, but the real kicker is the fact his film was destroyed on the impact of the fall. So not only does this mean this entire trip was totally pointless, but he wouldn't be able to complete his promise to his dead friend, donating its proceeds to the man's family to keep them financially stable. So when this happens, he SNAPS. He grabs the closest rifle he can find, and he starts kicking all forms of bug ass. It's not much, but to show that the character wasn't always a money-grubbing bastard who cared only about himself was a nice touch.
Jack Driscoll gets one during the stampede scene, where he—without breaking stride—shoulder checks a Venatosaur into the side of the canyon, triggering a chain reaction of dino falls behind the group and saving their lives.
The only worthwhile scene in "King Kong Lives" is also its Moment of Awesome, where Kong takes on an armed platoon of soldiers, (led by an almost "Ax-Crazy" colonel) taking machine gun fire and cannon shots to the chest as he wades through their numbers, either killing or scattering them all. He's dying, yet he's not gonna let up till his enemies are crushed and his mate is protected. He finished this up by crushing said colonel under his big fist.
Saving Kong in the 2005 game's alternate ending.
In the 2005 version, after Kong manhandles her a few times out of amusement, Ann rightfully snaps and tells the oversized ape to knock it off! It gets him to treat her with a great deal more respect after that point (after the obligatory temper tantrum, anyway).
In Son of Kong, Kiko's battle with the cave bear is a grueling affair for both parties that finally ends with Kiko arms himself with a club and beats the bear into submission. It may not be the T-Rex battle from the original, but it's still an amazing watch.