The first consoles
My first console was one of the cassette ones. It was so long ago I don't exactly remember which one it was.
The first real console that I remember asking my parents to buy for me, specifically with the name, was the PS2. The first games I played on it were XIII and Freedom Fighters, and then the third one was Need for Speed Underground 2.
The PS3
I care, genuinely and slightly ridiculously, about how hugging a fat 60GB backwards compatible PS3 makes me feel. Like hugging a friend from childhood. That is the actual reason. Everything commercial here exists so that reason can keep going.
It might not have been the first console that I had as a child, but it was the most difficult one I had to convince my parents for. It was super expensive when it came out and it was a huge jump from the PS2. I think I was 13 when I got it, and the first game I played on it was MotorStorm. I still like MotorStorm.
Years later
As I grew older I found myself in situations where I had less and less time to play games, but more and more time and a little bit of money to actually buy them. That was something I was not able to do when I was growing up.