I'm surprised that it's not on the roadmap, so here it goes.
Anyone who has played a number of "multiplayer" mobile games has probably seen push notifications coming by. In short, if two or more players are involved in a (usually turn-based) game, they may receive a notification on their phone / tablet that it is their turn in the game, or that they must perform some action in the game (think WordFeud, for example). Notifications appear even if the game is not running. Tapping the notification starts the game, or switches to the game.
For Android, push notifications for games may be supported using Google Cloud Messaging, which is free. Apple has a comparable service which is paid. I haven't investigated such services for Windows Phone and RT.
Note that notifications aren't supported using the multiplayer functionality we already have in GameMaker - the difference is that the app may actually not be running.
I'm aware that this feature may require players to grant an extra permission to the game. I'm also aware that we are required to build our own server to issue such notifications to cloud messaging services. But it would be extremely cool to have.