Mary Gascho has a persistent pest. After upgrading her iPhone, “I am constantly getting an ‘approval request sent’ popup.” The message tells her that if she wants to keep using iCloud, she needs to approve that from another device that is using the same iCloud account.
This appears to be related to iCloud Keychain based on reports from other people having the persistent problem. My colleague Joe Kissell wrote a thorough rundown in 2013 on using iCloud Keychain, including approving access from one device for another, and that’s remained largely unchanged.