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?
Asheron's Call was probably the single most defining game of my childhood. The game itself doesn't really hold up to today's standards in most regards, but it had so many cool concepts, some of which have never really been explored since. I loved the way that "quests" were more organic. You didn't go up to someone with a mark over their head and add a task to your log. You would have to just pay attention to stuff that npcs talked about and infer from that what might be going on. The monthly updates and gm interaction were just a completely different experience from modern MMOs. It also had the advantage of being from a time before everything was mapped in detail on the internet. Maggie The Jackcat existed for some stuff, but it was more of a blog than a resource like wowhead or the like. The game just felt like an adventure, and that type of experience can't really be recreated today.
On the single player side, definitely Panzer Dragoon Saga. I played through it again a few years ago and the story still holds up pretty well. I loved the exploration, and the customization of the dragons. There was just so much to do, and it kept me busy for months.
I agree with you on Asheron's Call. I can't say it was my top pick but I played it quite a bit and thought it was quite an interesting one. I belonged to a guild of some sort. My roommate got hooked on it too and so the social aspect of it really made it for me.