iCloud Backup Keeps Asking to Verify Password (over and over again)

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After having an issue where the "Other" storage on my iPhone grew to over 5 GB and doing a restore to fix that problem I had another problem with my iPhone where the automated backup to iCloud stopped working.  I'd plug in my phone to charge and when I picked it up later, I'd have a notice saying something like "iCloud backup requires that you verify your password".  I'd enter my password and it would be accepted fine.  But, apparently the backup was never happening.  The really weird thing is iTunes was showing my phone was last backed up this morning.  But, the iPhone was showing it was last backed up last week (when I did the restore).  I don't know which was true.  After a week of this and not finding any answers by Googling the problem, I got fed up.

I called Apple Support and I did some research on steps to take.  I then did a better Google search and found a couple of simpler steps I could have tried first.  Here are the steps that have worked for other people and the one Apple Support gave me.  They are in order of easiest to hardest first (Apple started me with the hardest).  None are very hard and after doing this your phone should be running backups smoothly again.

First

1.) Go to Settings-->iCloud-->Storage and Backup

2.) Click on Back Up Now to force a backup from your phone.  This seems to work for many people and is simply forcing a manual backup from your phone.

Second

If that doesn't work do step 1 above and turn off and then back on "iCloud Backup".  This might be enough to reset the iCloud backup app.

Last ditch effort (this is what Apple Support had me do)

If that doesnt work go to Settings-->iCloud and then Delete your account. Before doing this make sure you remember your ID and password because you'll have to re-enter them.  Choose the option to leave the data on your phone.

Then, add your iCloud account back.  Again go to Settings-->iCloud.  When prompted, you'll choose the option to merge your data with iCloud.  You'll have to go back in and turn Backup on again and turn PhotoStream back on again (assuming you want PhotoStream on). After doing this I forced a manual backup (see First above) and now my iPhone and iTunes both show the same time for the last backup to iCloud.