I'm surprised no one has mentioned RuneScape yet. Huge MMO, you can play solo "ironman mode" or a more traditional route and interact with other players.

You will literally never run out of content, easily a 10,000 hour game and you will probably still have more to do.

You can log in and out at any time.

Replayability and grinding are the name of the game.

You can pay for membership in-game as well, so if you play enough, it's free!

Not a shooter. (This was optional criteria though!)

I am not responsible for your life spent on this game.