WK's thread got me thinking on some games that start out weird. This isn't just about opening animations, but the beginning in general. Think about the first hour or so.
My top two:
FFX-2. I didn't read a whole lot about it, and I was expecting something like Final Fantasy X. Imagine me and my little bro's surprise when we pop it in to find Yuna doing a concert followed by a Charlie's Angels style introduction of our three heroines.
Persona 4. I knew TVs and weather would be relevant to the plot, but it was pretty wtf when the protagonist almost got sucked into his television.