I used to like going to Miniclips. I am also a big fan of Cartoon network. I enjoy playing games that challenge me to do better each time I play them. I also like to play cooking and baking games online to. Miniclips offers a good selection of games to choose from. As does Cartoon Network. I am addicted to a game on Cartoon Network where you have to fight off evil toys on Christmas. It's a great way to blow off some steam.
I used to play Y8 when I was a kid. And years after, I found newgrounds and there are some games that are great.
Miniclip is probably my favorite, you can find all sorts of browser-based games there and I really like playing them. There are so many genres it's pretty crazy, you can never get bored.
I don't really use such sites anymore, but back then I really liked Neopets, since there was an added pet-raising aspect and social aspect aside from the game arcade.
I used to go to NewGrounds a lot but eventually lost interest. Now that I remember it though I'd love to get to okay again but sadly my laptop just broke so I can't. When I get a new one I'll definitely check it out again and see what I've been missing for the past decade.
I think runescape was always my favorite browser based game, although these days it has evolved beyond that.
Well to be honest, like a lot of the other users here, I have not been to a site like this for some time now. Thanks to @Wubwub for reminding of NewGrounds, and from what I do remember that one was pretty naughty. I think that was the one that all the shooting games and were pretty offensive. I also went to addictinggames.com or something like that, but again it has been years.