Defeating Primal Dialga as two unevolved Basic Pokémon.
One of the special episodes available in Explorers of Sky allows you to play as Grovyle. And in the end, you team up with a reformed Dusknoir and Celebi in order to fight Primal Dialga, with epic music playing in the background.
Another special episode, "Bidoof's Wish", has the moment when Bidoof is alone, beaten up by Snover and his rogues Bagon and Gligar, and about to be robbed, when suddenly the entire Guild appears and thoroughly curb-stomps the robbers in an epic battle for daring to hurt him.
Even better if you somehow manage to take them all on and win. The Battle Didn't Count is in full effect, but still.
The Atoner Drowzee sending the heroes into Azurill's nightmare was cool. What's even cooler? Saving them from The Big BadDarkrai himself. Ridiculously awesome when you realize that his moment of suddenly showing up saved the heroes and scared Darkrai away.
On top of that, he (along with a few others) were perfectly willing to go with you to Dark Crater to defeat Darkrai. It wasn't to be, but the fact remains that many people were ready to help out. Also a Heartwarming Moment.
The special episode "Today's 'Oh My Gosh'" is rather underrated, for the moment when Sunflora apprehends the "invincible" Haunter:
Sunflora: I have... I have proof that you're not invincible!
While ineffectual, the Sableye unhesitatingly attacking Dialga when Dusknoir is threatened in "In the Future of Darkness".
The partner's development as a whole. Starting off too scared to even walk onto the guild's front door. Slowly develops more and more until s/he becomes badass enough to (with the main protagonist) defeat two (and more) "Legendaries" (By which are minor GODS in the Mystery Dungeon universe) without even evolving If the game was any longer they'd probably fight Arceus himself and win. the hero was arguably already a Badass before the events of the game and it just came naturally, as the hero and the partner are kind of the Mario and Luigi of the team
The so-called graduation test after the main storyline. Taking on your Guildmaster and the rest of your guildmates—sans Bidoof is one thing, but doing so with only your partner as your sole ally? You're definitely showing your mettle by defeating them, and proving to the game that you're ready for the harder dungeons in the epilogue, since it takes preparation and strategy to take ALL of them down in one big boss battle.