I'll start:
I could never choose a single game, but some of my favorite games that I played as a child are Rollercoaster Tycoon 1 & 2, The Sims 1 & 2, Medal of Honor Allied Assault, Runescape 2 ("OSRS") and GTA San Andreas.
The RCT and Sims games gave me a lot of freedom, while making it hard to screw up. It was so cool that I could design my own house or amusement park. I loved spending hours doing just that. I also learned a lot about living life, managing people and things like economics.
Medal of Honor Allied Assault was my favorite shooter in that time. It very well might be my first proper FPS. The atmospheric story-driven campaign drew me in a lot. The music and missions gave some very intense moments and the online multiplayer was absolutely amazing. Rifle-only battles, freeze-tag or a regular (T)DM were a blast!
Runescape is one of those games that I never really get tired of. As a child I only played as a free user, while being impressed by every member I saw. I loved the atmosphere, the people that I met and the progression of my character. I went on adventures in the wilderness with classmates or went mining for hours to make some money.
I can still get drawn into this game and really feel like I'm on MY adventure, where anything might happen. There are not many games that have this effect on me, so intensely.
This game also learned me a LOT about life. I learned about having to work for getting a result, I learned about economics and how you can use markets to make some money (this was long before the Grand Exchange). I also learned to watch out for ill-intended people: I stopped playing for a long time when 11 year old me got scammed out of my gold-trimmed black armor that I had been saving up for for a long time.
Lastly GTA SA made me feel in love with the GTA series. I already loved previous games as I had played a lot of GTA 2 and a little bit of GTA 3. But San Andreas was on another level. The huge feeling map, the intriguing story and all the thing that I could do blew me away.
I loved learning about the lore/backstories of the characters and even joined a GTA-related forum which opened up even more to me. I stayed a big fan of GTA and Rockstar Games up untill GTA 4 and bought all theirs games, often multiple times on multiple platforms. GTA 5 was fun to me, but it never really got to me like the previous entries did. I think this is partly because I really enjoy the stories and characters of the previous games, and the (admittedly interesting) choice to use three switchable protagonist resulted in character development that wasn't as deep and refined as games like GTA SA or GTA IV. But San Andreas... Man, I love that game!
Now I'm curious about the games that you loved playing during your childhood! What made them so special to you?