Starting with my favorite:
Perfectly sets up my second-favorite Zelda game. This is a story of mythical proportions, and this manages to summarize a fair amount of history in a simplistic but ambitious way.
What I'm saying, I guess, is that this is as iconic and perfect in my book as the Star Wars opening crawl or the Lord of the Rings intro.
As with The Wind Waker, this sets the tone well.
Shenmue was about leaving the comfort of home to set out on a grand adventure. In the original game, you really do inhabit Ryo's life, and the people who surround him become your friends. Yokosuka becomes your home.
Shenmue 2 is entirely different. Hong Kong has a magical, foreboding atmosphere, and feels far more surreal than the rural Japan of the first game. The opening, therefore, is fittingly jarring. There's an openness, an unveiling. Ryo's whole perception of his heritage, his destiny, and the world is about to change--and the opening sets you on that course.
Shenmue 2's opening tells you that what came before was just a prologue. The story starts here.