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?
Sonic and Knuckles- This was probably one of the first games I ever played, which is what makes it special to me. The game and the Genesis I played it on were hand-me-downs from a cousin. I never managed to actually get very far, but for a while, it was my go-to game to play after school.
Spyro the Dragon Trilogy- A few years later I got a PS1 for Christmas and Spyro the Dragon was the game I got with it. Like S&K for a while it was the only PS1 game I had so I was constantly playing it and replaying it. Spyro was the first video game franchise I actively followed and begged my parents for. Then Enter the Dragonfly happened, but at least by then, I had Ratchet and Clank.
Elder Scrolls III Morrowind- This was my first real RPG and open-world game. Go anywhere and do anything has kind of become a Bethesda marketing meme, but my twelve-year-old self was floored by the fact that I could actually do that. It was the first game I had played that let me just ignore what the game wanted and let me wander around and make my own fun, which amounted to me wandering around and reading every book I could get my hands on to learn about the world.
Persona 4- Not sure if this technically counts as a childhood game since I was around seventeen or eighteen when I played it. For some reason up until P4 I had convinced myself that I didn't like JRPGS that much. I had played a few growing up, but none of them were really my favorites. Honestly, I probably would never have picked it up if I hadn't seen gameplay on youtube. I had sort of stereotyped JRPGs as all looking like Final Fantasy or something like that and Persona was just so different that I had to pick it up. Ended up loving it and the Megaten franchise, which got me into JRPGs in general.