Q: Family Sharing Approval of App keeps Looping

Did you ever find a solution to this?  I am having the exact same issue.

iMac, OSX Yosemite (10.10.1)

No, did not find a solution, but the looping stopped - it may have been on one set of requests - I'm not really sure.

I am having other issues with Family Sharing request - notifications don't appear on all devices and once approved they app is not loading on requestors device etc.   Great concept but its harder than it needs to be at the moment.

Same problem.  Completely idiotic.

Getting exactly the same. Daughter's iPad made the request. I get the approval on my iPad and MBP (but NOT my iPhone?), and on both iPad and MBP, the password request just loops around and around. I know the password is correct - if you type a wrong password it will tell you immediately.

So how does my daughter no get this app? there's no other way.

I've only turned on family sharing for an Apple Music family account. If it doesn't work then I'll turn it off and use Spotify instead.

Same issue here.  Loops on all devices, iphone, ipad, macbook pro.  If my son goes into the app store, finds a purchase which is listed already, he can load the app without prompting.  Loads fine.

Same - it all used to work, not now

Same here. Authentication failed, repeatedly Asker for password. The purchase becomes approved though, so I just cancel at the 3rd password prompt to be sure.

Almost two years later .. Same problem here. And only sometimes. Only some app's maybe? Dunno..

I struggled for hours with the Family Sharing Approval process repeatedly asking for my iCloud password.  I eventually contacted Apple Support and was about to do an iPhone reset (wiping everything off the phone and thus essentially starting with the equivalent of a new clean iphone).  I had tried everything up to that point:
-closing all apps and powering the phone off/on
-logging out of iCloud and restarting the phone

-changing my iCloud password
-re-entering all my personal and charge card data, etc, etc.

I backed up the phone and was about to wipe it clean, when the support representative said "let's try one last thing, go into 'Settings>iTunes & App Stores' and 'Sign out', then sign back in again. "
That solved the problem!

I went to 'Notifications' found the request for Approval from my child, I 'Approved' the purchases, was asked for my iCloud password (once only) and the purchase transaction was processed properly.

I had assumed up to that point that logging out of iCloud vs. iTunes would have been one and the same thing since they both share a common log in and access all the same Apple ID server information.  However, apparently this is not true.  Some people have claimed that this problem can be solved by logging out of iCloud... and since the sign-in message that is the cause of this problem states you need to enter your "iCloud password", this logically makes sense.  However the sign in message is a 'red-herring'.  That message is a programming dialog error.  It should read 'Enter your iTunes ID password'.
It is not the iCloud password or ID that is the problem, it is in fact the iTunes App that is stuck in a login loop. 

Simply 'sign-out' ('Settings' > 'iTunes & App Stores' > click on your 'Apple ID > 'Sign Out') and you can then 'sign-in' to iTunes again using your Apple ID and your problem should be solved.

The solution is hidden at Request and make purchases with Ask to Buy - Apple Support:

If you decline the request or don't make the purchase, it will be deleted after 24 hours, and the child will need to make the request again.

So what's going on here is that you're trying to approve an expired request, and it's failing, over and over, without notice or explanation.