I had a list of some flash games I used to obsess over, but I can't find it right now. I'll throw Sonny 2 as my recommendation, I beat that game many times and I still come back to it every now and then. It's a top-tier RPG.
I would also recommend Gemcraft and Bloons TD, but they have premium sequels now that are way better than the web games.
Slither.io is definitely my most played browsergame. It gets way more fun with a mod that lets you zoom out a little, as the bigger you get the worse the experience normally gets..
I remember spending a lot of time playing browser games from Nitrome.
Mutiny, that cave runner game, Dirk Valentine, Ice Breaker, Bad Ice, etcetera.
I also played a lot of the Bloons series. Hell, when I found out about BTD5 flash, I thought it was some fake fan made game that would give me a virus.
For me this is a hard toss up between Pizza King too and the Submachine series, the Submachine series is an amazing escape/puzzle style series that has a really good story
and Pizza King 2 is the best management style Pizzeria Simulator game that I have found to date, many of those management style games focus more on the cooking aspect than the management aspect but this one takes the cake with being an actual management Sim instead of a cooking
cookie clicker, that chromium dino game.
I think I also did dark room at some point.
that pokemon browser game.
There's also a taiko game out there. (haven't checked lately)
I spent a ton of time in Kingdom of Loathing. I kept an old laptop that still had a flash enabled browser for a few months to play Bloons Tower Defense 4.
There was this game, I forget the name of it, but you had to build and drive little vehicles to overcome challenges. It was technically amazing for a Flash game, and I'd love to have it back.
I've spent a truly mind blowing amount of time on MissionChief. They don't give exact stats, but I'd say I've played at least 2k hours over the last 4 years
Amazed no one has mentioned Antimatter Dimensions. It's now on Steam, but was originally on a few sites (I played it on Kongregate). Fantastic, and now with the latest update and the 'aero' skin and hidden games, it's multiple games (Including the original Paperclips) inside one :D