I don't play a whole lot of games either! The most obvious example of an intro-less story driven game that comes to mind is Undertale, where you get to walk around immediately after choosing a name. Of course, the next few minutes incorporate the game mechanics so it's sort of like a tutorial; I don't know what exactly is considered an intro. There is also the title screen story thing, which I don't know whether it counts. I mean in this one you still have some control of the actions you're choosing in the battle, right? In general I think it's more important what actually happens vs the extent to which the player has control.