I had a problem where System Preferences would alert me that I had to "Update Apple ID Settings", and that some iCloud features would not work until I did. Clicking on the alert would bring up a dialog asking me to enter my Apple ID password, but doing so would just bring up the original alert again.

Apparently this is a widespread problem, both on the Macintosh and on the iPhone. A google search on the message brings up lots of other people asking about the same problem. Most of the replies relate to the iPhone. None of the Mac-related replies were helpful.

I did find the solution for the Macintosh. I posted it on the Apple forums, but my solution there was deleted, apparently because it was a solution and not a question. (I couldn't post as a reply to any of the existing threads there because they were too old. As in, more than a few weeks since the question was asked. Come on, Apple!)

Anyway, the solution is to go to Terminal and copy/paste the following command:

defaults delete com.apple.systempreferences AttentionPrefBundleIDs

That doesn't make the message go away, but does allow System Preferences to break out of its loop. Enter your Apple ID password one more time, and it'll progress to a new alert asking you to enter your login password. Do that, and the condition i is resolved.