Hi John
Sometimes very little. I tend to do the in-game hud with components and systems, as in the Asteroids example. However, sometimes there are lots of menu systems and data services that are not part of the gameplay but are rather systems used to introduce and configure the game and show high score tables and the like after the game is complete. In this case I might use an MVC architecture to present these menus and just build the core game with Ash. The menus will configure and launch the game, then sit back and wait for the game to finish.
For example, in Stick Tennis, a tennis match is driven entirely by Ash, including the hud, pause screen and scorecard. But all the menus displaying the different game modes and letting the player select which match to play are outside of Ash and use a simple MVC architecture.
Richard