As Jolteon said there can be two meanings for this question: being too young to have reached the time a game was released or relatively popular yet still having played it; and being too young for the demographic of a certain game you've played. I don't mind whichever was meant so I'll answer both.
As for the first one, I don't think such a thing exists. Like most of you said you're not really limited to playing the games made after you were born, as much as that was the norm. But to stick to that question, I have played a lot of NES games like Super Mario Bros., Ice Climber and Balloon Fight back then even though the PSX and even the PS2 were more in-time with my "conscious" age. I suppose we got the older consoles way past their popularity. Even in emulators, I still play games way older than me.
Me being too young to have played games from a certain demographic though? I have experienced that. A childhood friend of mine a few years older than me enjoyed violence way more than me, even before violence was evidently prolific in media like it seemingly is now. So at around eight years old during pretend trips to an arcade of sorts, I've played Grand Theft Auto: Vice City in another house, obviously not a very appropriate game to play for kids like us. And I couldn't say it was something my parents were happy about.