Pssh. All you people with your Gameboy games or DS games or NES games. Let's take things back even farther, to a system released in 1977 - the Atari 2600.
Honestly, I'm not old enough to have been around when that was released, but my dad was. He had it when I was a young boy, probably around 5 or 6, and the first game that really got me hooked was "Spiderman" for the Atari 2600.
Any time my dad let us play, that was the game I wanted to play. Unfortunately, the controllers for the Atari broke after a couple years, and they would've been expensive to replace, so we never got new ones. Though knowing what I do now, I wish we had.
If not that, the next game I would point to would be Teenage Mutant Ninja Turtles 2 for NES. I never owned a NES, but I had a friend that did, and I remember that we always played that game whenever I came over. We barely got past the first boss if we ever did, and my friend got bored of playing long before I did, so I'd reluctantly quit after a run or two. I think that was the first game I could say I was addicted to, if I were to go that far. Then again, I hadn't really played many video games at that point as we owned no "functioning" systems anymore, so it was kind of like a treat I only got every so often. While Spiderman was the first video game I really enjoyed, TMNT 2 was the game that made me want my own system, though that wouldn't happen until around 3rd or 4th grade, when my parents bought us a used PS2 as a surprise. And it was the best surprise for me at that point. After that, things sorta just continued from there.