Chrono Trigger - ...I'm sorry, but there is nothing better than the fights against Ozzy. Freezing yourself is smart, but not when you're standing over a trap door, which is suddenly opened by the 'good guys' or a stray cat. Also, some of the optional endings are just too funny...Like the one where Marle starts croaking and hopping around like a certain frog-ish knight..
Breath of Fire 2 - The part where Sten makes Ryu 'Disappear' and is explaining things to Nina and Katt...When Ryu suddenly pokes his head out of the ground.
The funniest Nintendo moment for me was when Groose followed Link down to the surface in The Legend of Zelda: Skyward Sword. I loved the part where he saw the tiny birds and freaked the hell out.
I think cringe inducing when Shrunk tells most of his jokes.
@Oriana: The Groose is a boss who was clearly the main character of Skyward Sword!
As for hilarious moments, Gregor from Fire Emblem Awakening had some pretty golden lines and so did the Scramble DLCs. "How much you pay Gregor? .... That was joke."
I can't think of a single moment but the whole Warioware series for me. The microgames being thrown at you one after another and the absurdity of some of them makes them funny. I always found the happy mask salesman more creepy than funny.
Also made by the same team, Rhythm Heaven games are funny in the weirdness of them.