In the early game, you eventually discover that the mysterious bleak place you ended up is a really, really Bad Future which prompt your teamates to do something about it. If you decide against it, Lucca will ask Chrono if he's really willing to chicken out in front of Marle. Chrono's reaction is priceless.
Mom: No buts! The proper response is "Yes, ma'am!"
Ayla: Crono strong... Crono mom strong, too!
The fight with Golem Overlord.He counts down like other bosses do before their ultimate attacks, but then he succumbs to his fear of heights, freezes up, and eventually ends up fleeing.
In Ozzie's fort, Ozzie summons two monsters on a conveyor belt to fight the heroes. The battle music starts, then both enemies move along the conveyor belt and fall into pits at the end of the belts. The music abruptly stops and your team needs to take a moment to blink and let it all sink in.
The unique item Ozzie Pants provides good armor at the expense of inflicting Confuse on the wearer. What item slot do these crazypants take up? Headwear.
It goes one better. We're using the British definition of "pants" here. No wonder you're confused, you're wearing underwear on your head!
Yeah... And worse, they're Ozzie's underwear. Guess "Confuse" was only because "Nausea" wasn't a real status ailment...
Any encounter with Ozzie is hilarious, you fight him and after losing his minions, he says "Ozzie in a pickle (or also says in a jam) Outta here!" and runs away like a little boy. The final encounter is taking the comedy up to eleven after falling into some traps, you return to fight him, and a little kitten runs up, presses the switch and meows like "There you go!" and Ozzie falls into a hole in the floor.
Prison break from Guardia Castle, either scenario is pretty funny. If you wait for the execution, Lucca comes running in, and conks out the guards. If you escape, you get to the jailer, and he runs for help and is knocked out by Lucca. "This is what happens to fool who challenge me mighty Lucca!
Go up to talk to the unconscious jailer, you get five free elixirs!
The meeting between Ayla and Frog. "Big frog present? For Ayla eat?"
This is only in the SNES translation, but thanks to the replacing of alcohol with soda or soup, you get this hilarious bit of dialogue:
Man: Hey, bring some more soda!
Bartender: Sir, I think you've had enough sugar today.
Man: Pipe down! I'll tell you when I've had enough! Can't you see I love soda pop! Nothing like soda pop to quench your thirst!
The fact that Robo's theme is very similar "Never Gonna Give You Up". You get Rick Rolled every time you play the game!
Even better: the similarities are accidental. Yasunori Mitsuda had honestly never heard of Rick Astley when he was composing the music for this game.
The Rainbow Shell sidequest has a couple of funny moments when you discover Queen Leene's letter to Marle, depending on who's in your party.
After reading the rather touching letter, Marle remains quiet for a few moments. Cue Mood Whiplash when Ayla suddenly gropes Marle's breasts and wonders if she's really ready to "leave the nest" as she's "not big enough" for that yet. The scene ends with a bit of black comedy as Marle shields her chest from any further abuse while silently looking down, as if a bit depressed about the whole thing...
If Magus is in your party together with Crono (and Marle, obviously), he will be the one to discover the letter... and does so in a very deadpan manner, especially considering the contents of the letter:
Magus: There's a letter here. Shall we burn it?
Ozzie accuses Magus of using him (when actually it was the other way around) and Magus gets a good comeback
Ozzie: Magus, you lied to me when you said you wanted to create a world of evil! You used me
During the party in the prehistoric times, if you have Lucca in your party, she'll end up getting completely smashed.
In the Developer's Room ending, one of the rooms you explore is a prison cell, where you find a man named Hasui. Talking to him has him comment not to bother him, because he's "in the middle of something good". The sound effects when you close the message window imply he's masturbating.