There is no technical limitation to an app engaging in very nefarious activity. For example an app could modify the DOM on the sign-in screen to grab your password after you enter it. Of course the accountability of MITM and security reviews might mitigate this risk, but passwords are generally only a few bytes and could easily be obfuscated and passed surreptitiously over the wire.
I've seen a (quite popular) app implement this for facebook invitations, which should only have a limit of 50 friends, to secretly (in the background, without any action on your behalf other than logging in, or perhaps pressing a "continue" button) invite every one of your friends to download the app. Since facebook does not notify you when you send an inviation, you would not even know the app did this unless one of your friends who you invited asked you about it.
[0] https://news.ycombinator.com/item?id=11637209 (thanks molecule)