How do i stop my ios and osx devices continuously asking me to enter my appleID password?

My various apple devices and apps continuously ask me to confirm (enter) my AppleID password. By continuously, i mean at least a few times every day on each and every device. Sometimes, on Podcast on iOS, or iTunes on OSX, it will keep asking me every 15 seconds. Sometimes on iOS it will let me get halfway through before the dialog asking me goes away and comes back anew (and empty entry line) 5 seconds later.

I have tried resets, typing in the credentials anew at the appleID website, etc., but it is not going away. Any suggestions? (and yes - i know and have checked the password, which is accepted frequently along the way)

This seems to have been building over the last 6 months, but recently (last week or four) it's been nuts. Lots of resets along the way to try and fix it, all to nought.