Was gonna post the FF6 intro myself, haha. Such a great game all around.
I really enjoyed the Pokemon Crystal intro. The cinematic is very short, but so mysterious, and Suicune is the bestest boyo. I watched it way too much.
Sticking with the nostalgic titles, here's Dragon Warrior 3. A longer intro with a big setup for the story, and you even get to hear one of the best fight songs ever.