Computer games were an enormous part of my childhood, so I could list a
very large amount of them just off the top of my head.
1) Toontown - This is the biggest game that has ever taken over my life. From approximately 2007 - 2011, I played that game night and day. I took a (forced) break until 2013, as my dad deleted my account and blocked the website so I could not reach it. I still play pretty frequently on Toontown Rewritten. I don't think I will fully be able to quit this game until I can no longer play it.
2) Club Penguin - I played Club Penguin pretty frequently from 2010 - 2012. I had a year-long membership in 2011, and I played this game frequently through a shared account with my brother. I look back fondly on playing the spy missions, card jitsu, and eventually, once they released, collecting badges on our account.
3) Wizard101! This is the second game that I had a major obsession with. I began playing in 2012, and I didn't really know what I was doing. All of the friends I made on my first account I had lost contact with completely, as I was too young to remember and didn't bother contacting them outside of game. However, this changed in 2013; I made a second account and befriended a very large amount of people. I look back extremely fondly on the memories I created with friends from 2013-2014 on this game. It was a very dramatic time with lots of petty fighting and online dating, but I still loved it. I still am very close with my friend Dani that I met in 2013. I still play this game on and off, but it's not fun as it was back in the day, especially when most people have quit (that I knew) by now.
4) Maplestory - I'm surprised nobody has mentioned this yet! I played this game pretty hardcore back in my freshman year of high school. I bonded with my friend Mirie through this game, and I am going to download Maplestory 2 soon so that we can play that together!
5) Roblox - This was a huge childhood game for me. Beginning in 2012, I played this game very frequently until 2013 or 2014. I used to have Outrageous Builders Club and I made various friends that I do not have contact with anymore. I still play this every now and then, but it isn't as fun as it used to be.
Those are the major ones, but here are a long list of more minor ones that I used to spend quite a lot of time on: Poptropica, FaceBook games (Farmville, Pet Society, Wild Ones, etc.), girlsgogames, Big Fat Awesome House Party, Fusionfall, Postopia, Millsberry, Moshi Monsters, PetPetPark, Bearville, StarDoll, and I'm sure there are others that I just can't think of off the top of my head.