The Pokemon franchise, preferably FireRed/LeafGreen version and Pokemon X (the only Pokemon games I played, really)
The franchise is super addicting and hard to put down when you are playing it. I can play Pokemon X for the whole day and not even notice it. It's just something that triggers in the game that makes it so addicting. I'm not sure what it is! It all started with LeafGreen that my mother gave to me for my birthday (later coming FireRed the next year). I got hooked to those games, and FireRed became the most defeated one, but I couldn't pass the Elite Four. I then broke my fourth GameBoy Advanced SP charger. I couldn't play them anymore. Didn't play Pokemon for years, until I got Pokemon X for my birthday more than a week ago. Now I'm addicted again!
I could also mention the Kirby franchise. I had Kirby and the Amazing Mirror when I was 5 for the same system. Loved the pink puff guy. Kirby's my childhood game. I also had Kirby Air Ride, another amazingly fun game, and I currently have Kirby Mass Attack.