Shoutbox

You could always put Messenger into boss protection mode, so it does nothing.
Alternatively if you wish to stop just Messenger toasts when you play games, by when you're in the main messenger window going to Tools > Options > and enable Show me as "Busy" and block my alerts when I'm running a full-screen program, such as a slideshow presentation". Still a conversation window will come up when you're in games, but on most games you shouldn't notice anything.