Progress Quest is the original 0-player MMORPG. It was designed in 2002 as a parody of the gameplay you know and love, with “fire and forget” convenience that went beyond auto-attack to auto-everything. Turn it on, create a character, and the game takes it from there. There is not gameplay as such, but it is a brilliant piece of work and strangely hypnotic.
As with most things, there is actually a genre of these by now. Kongregate has an idle game category. Epic Combo is notionally amusing, and Farm of Souls is more of an idle RTS game (with just peons). I’m tempted to fiddle with a few of these, but Kongregate is currently promoting Anti-Idle, which has enough little things going on at once to actually make it a game. It has its version of Progress Quest that you can play interactively instead of idling. It has a mini-FarmVille, a lousy Mario Kart, a collectible card game that doesn’t look very good but probably is not the worst on the site, a variety of mini-games, fishing, and some other things I have yet to sift through. It also has its own quests and achievements built in.
At worst, the gameplay is no worse than things you have paid to do (mine in EVE, farm almost anything). At best, well, it’s not a lot better than that anyway. When social media games were having their heyday, I found some of them interesting if I played 5 or 6 at once. The aggregate can involve interesting resource and attention allocation. Most of that seems built in here, plus your equivalent of offline skill training.