As others have mentioned, the culprit is DeepScreen. It's especially bad if you have Avast set to Silent/gaming mode because then you can get unexpected/bizarre behavior and have no idea why. (I spent an entire day debugging the installer for a program I'm developing because it suddenly started running twice every single time I launched it--DeepScreen is very forgetful--and only succeeding on the the second run.)
Going under Settings, Update, Details, you can turn off popups for the majority of them--the annoying update-related popups, which should make it easier to work undisturbed, play video games, etc. But DeepScreen popups still pop up!
CANNOT DISABLE DEEPSCREEN: I've tried disabling DeepScreen entirely (under Settings, Antivirus), since it's so disruptive, but this checkbox appears to have no effect.
Unfortunately, the only way I currently have of disabling DeepScreen (in Avast 2014.9.0.2021) is to disable all shields! And I'm not even sure that this is working, since I sometimes get DeepScreen popups even with all shields disabled. Avast can't be prevented from starting with Windows during boot, so maybe uninstalling Avast is the only solution.
[Note: I'd appreciate upvotes if possible, so I can get enough rep to participate here at SU via comments. My rep from stackoverflow doesn't apply here, sadly. Those feeling extra generous could upvote 253935 too... :)]