A bit difficult to choose one. Pokemon is a good contendor, and likely the winner when I think about how much of my life has been spent playing Pokemon games!
I've played one or more games from each generation, at least a couple times. And even now as an adult, I get that tingly excitement anticipating a new release and that sense of adventure when I start playing it.
Chrono Trigger has always been one of my favorite as well. Maybe I'm a sucker for 16-bit pixel art, but I love the colors and graphics in the game. And the music is something else; still one of my most beloved video game soundtracks to date.
Dragon Quest and Final Fantasy are honorable mentions. I'd probably like them both more if I were able to make myself play more of the games in those series all the way through. Regardless, Dragon Warrior Monsters is one of my favorite childhood memories! I always thought it's the crossover of DQ and Pokemon was way awesome, and spent years hoping for a fan translation of the PS1 remake.
Now there's a 3DS remake, and I'm eagerly awaiting a localization. Although part of what I loved about that game was the same appeal that Earthbound, Chrono Trigger, and Pokemon had: 2-dimensional pixel graphics from a top-down view. There's something inexplicable about that just can't be beat!