I'll probably think of some others later, but two that come to mind are the intros for Silent Hill and Ridge Racer Type 4. Each time I've played these games in the past (and occasionally still do), I've always sat through the intro video.
Silent Hill's intro pretty much establishes the mood for the game. I really like the opening theme, and it fits well with the video sequence. In a way, now that I think of it, it's almost like watching the videotape from "The Ring". If you remember in that movie, when Rachel watches the tape, it's basically events she encounters through her investivation. The opening of Silent Hill is like that. It used the most of the CG sequences from the game, and kind of made the intro like a preview of what you're about to go through in the game.
Even though the video quality is not very pleasing today, I still like watching the intro for R4. I like the theme song used for this game as well, and the video really kind of puts you in the mood for racing. Like you're ready to start a new day in your racing career as you start a new game.
Those two are pretty much my favorites, but I also like the intro from Chrono Cross, Parasite Eve, Gran Turismo 3, Amplitude, and a few others that I'm not thinking of now.