My favorite style, which I think maps well to functional use, is to have the entirety of game state (including random number seeds, etc.) be in one "world" data

My favorite style, which I think maps well to functional use, is to have the entirety of game state (including random number seeds, etc.) be in one "world" data structure, call it W. Then, you collect all your input (say, controller axis values) into another structure, I. So each simulation step takes you from W + I => W' which you hand off to the pre-renderer and the next simulation step. The pre-renderer will combine the game world state with static data (i.e. models and shaders) and produce game-state-free data that a renderer can display to the user. This is the basic framework I've used for twelve years when I've been able to implement it (i.e. not often at my day job). It's worked really well, but I actually have not applied it in a language that supports purely functional programming. So you can guess why I'm in this thread.