Uhura's whole scene with "Mr. Adventure", from her epic stink-eye when he comments that her career is "winding down" to pulling her phaser on him and ordering him to get in the closet, while declaring, "This is fantasy!"
Scotty explains that he's rigged the Enterprise for a skeleton crew:
Scott: A chimpanzee and two trainees could run her.
The janitor in the Spacedock lounge, just staring as the Enterprise powers up and leaves its berth.
"DANGER: SPACE DOORS ARE CLOSED."
Kirk: And... now, Mr. Scott.
Kirk: The doors, Mr. Scott!
Scott: Aye, sir, I'm workin' on it!
Just as Sulu reports clearing the doors, Scotty can be seen clearly relieved and thinking "Oh, thank God that's over..."
All of it building up to the Epic Fail of the shiny new Excelsior and its revolutionary drive, complete with sputtering engine noises.
GOOD MORNING CAPTAIN
The sound of the engine stalling really just makes the whole sequence. It's entirely Rule of Funny, but the laughs justify it 100%.
Even earlier than that, Bones' multiple failed attempts to pull a Vulcan Nerve Pinch on a Starfleet officer. Knowing that it's both him and Spock sharing the same mind during the scene makes it all the more hilarious with McCoy's over-the-top Oh Crap! face.
How about this exchange between Kirk and Kruge on the rapidly-disintegrating Genesis Planet:
Kirk: You fool! Look around you! The planet's destroying itself!
Kruge: Yes, exhilarating, isn't it?
Scotty finally lets Kirk know his secret to being a miracle worker.
Kirk: Mister Scott, do you always multiply your repair estimates by a factor of four?
Scotty: Of course, sir! How else can I keep my reputation as a miracle worker?
Kruge gets in one. After killing his previous gunner for destroying the Grissom (when he wanted prisoners), Kruge eyes his new gunner and says, slowly, with a Death Glare, "Gunner! Target engine only," then adds a dark "Understood?" as if to say, "Don't fuck this up, or ELSE." The gunner can only answer, "Understood clearly, my lord." If a Klingon ever shit his pants, this was the time.
The crew of big, tough Klingons who are clearly terrified of Kruge's pet.
As the Enterprise enters spacedock, the people in the cafeteria watch it through the window, stunned at the enormous damage it sustained from its battles with Khan. One of them is Janice Rand, who just shakes her head as if to say "Kirk, what have you been doing now?"
During the Enterprise's trip to Genesis, Bones works the science station on the bridge - at one point, Kirk asks for a report, which Bones gives—in Spock's voice. Kirk, Scotty, and Sulu (sorta in the background) turn to look at the good doctor with epic WTF?!? looks on their faces, and even Bones is surprised: