Phantasy Star Portable / Phantasy Star Portable 2 - the first series I ever actually grinded, and did literally every possible quest. Replayed the game a few times out of boredom even. Hands down my favourite PSP game ever.
LSD: Dream Emulator - a game I have literally played for countless hours on my phone, used a PS1 emulator. This game was insanely fascinating, and no game ever came close to making me that i, ever. I highly recommend it to anyone and everyone, hands down one of the most inspiring games. I had many dreams involving settings only available in this game, so it did truly live up to it's name.
Need For Speed: Most Wanted (2005) - a classic, the most interesting racing game EVER. I'd definitely say Underground 1 and 2 was great as well, but no racing game ever came close to how captivating the story in MW was. While after a replay a year or two ago, I did find certain aspects of the game repetetive and dull, I still believe it was the game that defined my childhood.