Kefka: '"Sickening! You all sound like chapters from a SELF-HELP BOOKLET!"
That, and the whole destroying the world scene.
Many RPGs let you fight the end boss with your entire party. Kefka lets you take more than that. He lets you form three parties to take him on. And, just like almost everyone in your party, he was born a normal man.
Everything he did during the siege of Doma. To Cyan's fans, that wound up being his Moral Event Horizon, of course.
Let's not forget him slaughtering countless Espers almost effortlessly and killing General Leo just shortly after more or less mind raping the guy. "How about a lil' Magitek mayhem!?"
When you re-encounter Sabin in the World of Ruin, he's holding up a burning mansion on his shoulders, and even has time for a casual chat, including the famous line "Ha, did you think a little thing like the end of the world was gonna do me in?"
Sabin gets another one during the ending, where he catches a large piece of rubble while Edgar operates on a door and then throws it away, saying that he didn't run away from his responsibility, but trained hard to support Edgar. "Now I know why I have these stupid muscles!" Note that if Sabin didn't rejoin your party, it takes 3 characters just to drag the same piece of rubble aside.
The second Humbaba fight is Terra's Moment Of Awesome. During the first fight with Humbaba, we find that the year's worth of emotional turmoil Terra has gone through, coupled with her inability to truly understand the emotion of love, has robbed her of her Esper powers, leaving her indescribably weak, and easy prey for Humbaba. After a while, the party comes back to defend Terra and the orphans from a second Humbaba attack. When a party member is blown away by Humbaba, Terra shows up, and without a word, goes Esper. The following fight has Terra in Esper Mode for the entire fight! In other words, the most awesome example of a Mama Bearever. It helps that it's immediately followed by a Heartwarming Moment and Terra re-joining the party with a souped-up Trance.
When all seems lost, and the world is dying, Setzer reveals one final hope: the Falcon, the long-sealed airship of his lost love, Darill. After walking down a long staircase deep in her tomb, throughout which Setzer narrates her tale, the scene switches back to the surface, and the golden ship rises from the ocean as "Searching For Friends" starts up, forever replacing the gloomy world map theme of the World of Ruin.
Gungho tricking Strago into thinking he was wounded in order to get him to finally locate and defeat Hidon once and for all to put the issue to rest.
The Opera - particularly, Aria de Mezzo Carattere. There's a reason why it's the picture for the franchise's Awesome Music page, and why it's considered one of the most famous scenes in the history of video games: it's just that damn good.
Just look at how many people have said the scene made them cry, and realize the people behind the game were able to elicit that kind of emotional response over a situation that's not even real within the game itself!
There's one near the beginning of the game: Edgar, king of the highly industrial Figaro Castle, wakes up to find his castle on fire and Kefka demanding Terra. When all seems hopeless, the two of them and Locke jump from the castle onto speeding chocobos on the ground. As if that wasn't enough, as they're racing away from the castle, Kefka can only watch as the castle itself sinks into the sand beneath him.
So you've just rescued Relm from the burning house in Thamasa, and it seems that you're trapped, right? Cue Shadow, swooping in and saving your bacon.
For more Shadow awesome: when Kefka has your party trapped and paralyzed on the Floating Continent, he swoops in, undoes the paralysis, traps Kefka, and tells you to run. Epic.
Any time you fight Kefka is a good time. It's so satisfying beating up the villain you Love to Hate.
Celes - who has recently rejoined your party after a good while - shows up on the Floating Continent. Kefka hands her a sword and says all of her treason will be forgiven if she kills her comrades. She mulls it over for a few seconds, then refuses in the most direct way possible - by stabbing Kefka instead. It doesn't kill him, but damn was it cool to watch.
During the Siege of Doma, Cyan steps out of the palace. Alone. He then proceeds to slaughter the entire Imperial detachment. Alone. (You only need to kill the commander, but who would consider passing this up?). Considering his Class/Job, he took lessons from another samurai.
Celes wakes up after a year in a coma, and after a (possible) suicide attempt, regains the will to live and starts Putting the Band Back Together, culminating in the rise of the Falcon listed above. Most everyone else was either roaming around or settled down in one spot without getting much done for a year, apparently accepting of the new state of affairs. But within a week, Celes has put a rudimentary party back together and motivated them to regain hope and have another crack at Kefka.