I think this has happened a few times to me. Good opening cutscenes can do a lot. Most of these came when I was a kid though. So, let's see...
FFVII. Might be the first Playstation game I saw played. My cousin owned it, and, from then on, I wanted a Playstation.
Abe's Oddyssey. Together with Exodus, ranks up as one of my favourite games of all time.
MediEvil. One of the funniest games I've ever played.
Kessen. First PS2 game I got. Came with the console. Made me fall in love with it instantly, and reignited my interest in strategy games and historical warfare.
GTA: San Andreas. I don't feel it's an exaggeration to call it one of the most immersive games of its time. Couldn't put it down from the second I picked it up.
Those are the ones I loved pretty much from the moment I selected 'New Game'. Other games didn't necessarily strike such a good opening note with me, or they just had bad tutorial levels, but still went on to be some of my most beloved games.