Be the youngest of four with two older brothers, the Super Nintendo was my first hand-me-down system and I played the hell out of it! They had so many games and back then the internet to me was like, Neopets, so I didn't know anything about them. I remember the most vital game for me was Chrono Trigger, it brought me into the world of heavily story driven RPGs. After that I played through all the final fantasys available on that system and then Secret of Mana.
Donkey Kong Country is worth a mention, I remember playing it with a friend into the wee-hours of the night trying to get past certain levels; it was just plain fun.
Also Royal Rumble, it was just so funny to do the actual Royal Rumble play and have people thrown everywhere and chairs coming out, haha.
Did anyone play Uniracers? I think that's what it was called. That was a weird one but very addicting.

My next hand-me-down system was Nintendo 64 and I the two Zelda games for that system will always hold a special place in my heart. I have played them so many times and never get sick of them. I have had many spirited debates and usually I'm alone in this but I love Majora's Mask more.

Also, Turok! Did anyone play Rage Wars? I remember playing with a friend and we never understood the whole chicken tag game. Butit was entertaining.

And of course Mario Kart 64, I remain undefeated because of the ridiculous amount of times I've played this game. That and Mario Party 2, which I still insist is the best of them all. Everything is on point and it's not too crazy like all the new ones, just simple board game style with fun easy mini games (except for that stupid Shy Guy race I can't win that one for the life of me).