Shadow of the Colossus and Final Fantasy VI. The reason I choice both is because of their respective final boss, amazing soundtrack for both bosses plus all the amazing things that happen in those games.

There's also Ocarina of Time, mostly because of the transition from young Link to adult Link, plus almost all the bosses and stories. 

Edit: there's also Metroid Prime, that game simple has the best bosses imo.

God there are so many games I want to choice :(

Nothing I've been a part of in gaming felt half as epic as taking down Ragnaros for the first time.  I'll never forget that.  Back then, that was a real badge of honor.  Even the first time you encounter him is just overwhelming, especially when he smashes your guild in a couple of minutes.  You just had this feeling of, "okay, we're doomed, but that was awesome."

With single-player games, I'm gonna go with something different.  I remember finally, finally beating Soul Edge at the end of Soul Blade.  I had to get really cheap and use Siegfried, too, because my favorite characters all seemed to suck against him (I still love you, Taki).  So, awesome, I beat him and Siegfried picks up the weapons, but then they graft to his hand and the next thing I know, I'm Nightmare, this massive, armored demon knight.  Back then, that felt like the coolest thing in the entire world.

mass effect 2 and 3

finalfantasy ix :)

star wars galaxies, no game can match how deep the game content was.

1. Halo 4
2. Mass Effect 2
3. Halo 3
4. Mass Effect 3
5. Ico

For its time, I wholeheartedly agree. Set the standard for "epicness" on consoles.

Other games for their time:

Zelda: OoT for its sense of scale and battles (particularly the final one).

Metroid Prime for its atmosphere, visuals and level design.

Bioshock for its atmosphere and storytelling.

Metal Gear Solid 4 for its cut-scene direction.

Shadow of the Colossus, followed by GOW 3 and the original GOW.

Xenoblade, duh. That game defines epic.

Epic battles.
Epic scale.
Epic music
Epic story.

Prime series had a lot of epic boss battles

Metroid Prime for the Omega Pirate / Ridely / Prime fights

MP2 for Dark Samus / Chykka / Quadraxis / Emperor Ing fights

MP3 for Mogenar (fuck this guy on Hypermode) / Omega Ridley / AU 313 fights