When trying to update existing free apps or install new ones, I get into a payment "Verification Required" loop
In the past, I had set my payment method to be a pre-paid Visa gift card, which I used to download several paid apps. That card no longer has a balance at this point. This seems to be what was causing my problem.

I was able to fix this in the Payment Method screen by adding a new credit card (I used my real one) and hitting "Done". At this point my apps started updating and I wasn't prompted to enter my Apple ID password again.

After I entered the valid credit card information, I went back in to the Payment Method screen and set it back to "None", which worked. I can now install and update free apps.