When Ash, Misty, and Brock run into the Pokemon Center because of the storm, they are standing in the doorway exclaiming about all the rain when Pikachu interrupts them by shaking his fur dry like a dog.
Everything Mew does, from messing with Meowth, to popping the bubble it saved Ash with.
While Mewtwo is gloating on how he is superior to Mew, Mew pays no attention and starts chasing its tail instead; at one point it even gives a look as if to say "Is this guy serious?"
At one point, Mew spots one of the big windmill-like turbines outside of Mewtwo's fortress. It flies up to one of the blades and sits on it. When the blade lifts, Mew falls onto the next one, then the next one and so on. Then there's an adorable close-up of it giggling in delight.
Mewtwo, the last character one would expect to have such a moment, has one in the original Japanese version; when attacked by Ash's Charizard, Mewtwo, not surprisingly, shrugs it off without a scratch and immediately quips "What a bad tempered Charizard". note Sadly, the English version goes with the much less effective "Your Charizard is poorly trained". A shame, especially since the English version could have easily delivered the original line and retained the humor.
Charizard was clearly insulted by Mewtwo's comment in both versions, so being called "poorly trained" or "bad tempered" despite being one of Ash's more powerful Pokémon at the time was a kick to the ego for him.
Prior to that, Ash's reaction to Charizard immediately attacking after he calls him out.
The face of Raymond (the 'pirate trainer') as he sends in his last three Pokemon at once against Ash.
Even better in the Japanese version, where he screams "OH MY GOD!". And even better than that, the guy who did his voice is the same guy who did the half-Japanese half-English ramblings for the incredibly silly "Pokémon Ieru Ka Na?" (the Japanese Pokérap).
A meta one, during the scene in the dubbed version where Ash gets Taken for Granite, Misty's VA at one point suggests to say "My bike!" to match the two-syllable movement Misty's mouth was making.
During the opening Team Rocket while spying on Ash and Co are envious of how well their eating. Jessie offers to cook up a meal only for Meowth to refuse saying the last time she cooked she wiped out eight of his nine lives!
From "Pikachu's Vacation," when Charizard falls butt-first onto Meowth, who's holding a bomb, which goes off right after Charizard lands on him. Charizard is unscathed but Meowth is left Ash Face'd in a crater once he gets up. Out of context, the whole thing is even funnier (or grosser) because it looks like the result of Charizard ripping a monstrous fart.
If you're a jerkass, or just into Black Comedy or noticing stuff in scenes everyone else finds sad, the fact that Pikachu being unable to revive the Taken for Granite Ash is hilarious, because it's about the only time in the entire series Pikachu's Electric moves do what they're supposed to do. Which is to say, nothing, because, save for the fossils, all Rock types at the time were part Ground.