Omega Boost:
The music, the FMV, it was just silly and oozed attitude. And it was god damn confusing. Some dudes go back in time to get the very first AI and you have to go back in time to stop them. The Pilot of the Omega Boost (I think he was even played by Steve Blum) is such a bad ass and he KNOWS he's hot shit. you get in the mecha and the mechanics all start cheering as you walk off to launch. And then there's the Jamacian mechanic who is like the pilots best friend and they're bantering and you have no idea what the hell they're saying because the damn awesome song is blaring away. It's awesome, it's epic.

Street Fighter x Tekken: yeah an odd one but the opening track and cinematic for the game are awesome.

Colony Wars: Vengeance: It lays out the setting completely. tells you what's going on, why you're fighting, and all the sides involved.

Xenogears:
It's confusing, it's weird, and it's an opening you won't understand until you're well into the game.

Starsiege: It's narrated by a child with a twisted poem to start. It feels like Warhammer 40k with mechs. Honestly Earthsiege/Starsiege/Tribes is one of those universes that really needs to be revisited and fleshed out.

Quake 3 Arena: Again just another opening that hypes you up to play. And the opening Animation with Sarge was pretty jaw dropping at the time. It was a VERY simple arena shooter but man was it a blast to play.