I'm still having this issue. Escalated to Level 2 AppleCare support and waiting on them to come back to me. The list of things I've tried without success is growing so I've updated my previous list below:
So far I've tried the following without success:
- Signing out of iCloud and iTunes & AppStore / Restarting phone / Signing back in to iCloud and iTunes and AppStore
- Selecting Settings>General>Reset> Reset All Settings
- Disabling iCloud Backup > Reboot > Enable iCloud Backup
- Deleting all Email/Calendar/Contacts accounts in Settings>General>Mail, Contacts, Calendar
- Restoring iPhone through iTunes. I've tried this with 2different backups. Both backups were created before the problem started in iOS 8.1
- Restored iPhone and set up as a new phone. Skipped all prompts to log in to iCloud during setup. Then added email accounts and logged into iCloud - After five minutes the popup started again
- Restored iPhone and set up as a new phone. Skipped all prompts to log in to iCloud during setup. Then added email accounts and LOGGED INTO A COMPLETELY DIFFERENT ICLOUD ACCOUNT. Same problem.
- So, that means that it can't be a problem with my data - Setting up as a new phone didn't resolve it
- Can't be a problem with my iCloud account - as the same problem happens with a different iCloud account
As I mentioned before, the only way I've found to stop the popup asking for iCloud Password (when I create a new mail or enable Airdrop for Contacts only) is to sign out of iCloud and leave it switched off.
Note: When I sign out of iCloud / Reboot the phone / Sign back into iCloud it can sometimes take 10 or 15 minutes before the popup will start to appear again, which makes it difficult to troubleshoot when you're on the phone to AppleCare. However, a quick way I've found to replicate the issue is to first enable airplane mode to cut all network connections - then try to send a new email. This causes the popup to immediately appear for me. Then it happens every time until I sign out of iCloud.