Ace Attorney series
All five games in this amazing series have epic final showdowns between the lead character and the ultimate villain. All of these moments are just loaded with awesomeness, especially since you are against all odds during these fights. The main thing that sets these games apart from others is that it makes you truly feel the pressure- you truly feel like you're in the midst of a desperate struggle, and the endings are the most gratifying experience in any game ever.
Heroes of Mana- Goddess of Darkness/Anise
Another amazing battle. The desperate struggle for the world's very existence, and the culmination of all the pain, suffering, and sacrifice up to this point. When you find out the true reasons behind the villain's actions, it just gets that much more emotional.
Professor Layton and the Unwound Future- Clive
Well, not technically a battle, per se, but it might as well be one. This was amazing, intense, emotional, and just plain epic all in one.
Sonic Adventure 2 Battle- Final Hazard
Aside from the horrendous name choice, this was another emotional and perfectly executed boss fight.
Pokemon Mystery Dungeon: Explorers of T/D/S- Primal Dialga
The most emotional fight in any game ever. You know that by defeating him, you will die, but you do so anyway. Not to mention the beautiful music all the while.
Mega Man Zero 2- Commander Elpizo
Your former leader and friend, driven mad by power and feelings of helplessness. Finally crosses the point of no return by killing Mega Man X. He tries to kill you himself, before you finally put an end to his destructive rampage, but not without feeling the pain of having to kill him.
Cave Story- Ballos
The perfect setting- the island is collapsing, being torn down by Ballos' uncontrollable magic, and you must kill him to finally end his rampage, or be killed yourself. It doesn't get more epic than this, especially after you finally uncover the last of the game's secrets while travelling through the Sacred Grounds to get to Ballos. The last area's only purpose is to set the story up for the final showdown.
Wind Waker- Ganondorf
See above reasons.
I Wanna Be The Guy- The Guy
This one is just epic. That is all.