These could be games that left a lasting impression on you, games that had stellar gameplay mechanics, characters that captivated you, games that you played tons of hours on, etc.
Mega Man X on the snes was my favorite for the console, the day I got to play the first time I managed to beat Chill Penguin, Spark Mandrill and Armored Armadillo. I couldn't for the life of me, for the next 5 or so hours, beat any other boss. That I played that long without any progress probably shows my dedication.
Donkey Kong Country 2 was my second fave. Never managed to "legally" get all 75 kremcoins, or beat all the 5 special stages back then. Hell, even getting to the final world was a challenge back then.
On the ps1, Mega Man Legends 1 and X5. I'm only counting the games I played when I was "a kid" (< 12yo). I still love most Mega Man games.
I personally maintain that the first Megaman X is the best game ever made. The controls are so tight, even modern developers have a hard time achieving something similar, the progression is straight up perfect, the story is great and the blend of easy/hard secrets allows for a lot of replayability for a game that old.