It's a tedious thing to do, but I'd make sure your contacts & calendar are backed up, get a list of your apps, then do a factory reset. Re-install your apps, fresh from the play store, a few at a time. If the popups happen again, then you will be able to narrow it down to those that you had most recently added back.
Be aware that it isn't just one app that's causing this either, and when you do find which ones are doing it, make sure to make a note somewhere. Also, if you let someone else play with your phone, make sure they know not to install those apps as well.