"macOS needs to repair your library to run applications"

Hello, Yale Linguist.

Sorry to know that you're having this issue with your wife's Mac. I think we can help.

First and foremost back up your wife's documents, files, photos and music – all her stuff. Time Machine is the best recommendation for that, but you can also use apps like CarbonCopyCloner or Super Duper!

Back up your Mac w Time Machine - Apple Support

Since she has accumulate so much stuff you will want to offload as much of her "stuff" as you can to an external drive to free up the space that macOS requires to do its thing.

Then use Recovery mode to reinstall macOS Mojave, erasing the drive in the process.

How to reinstall macOS from macOS Recovery - Apple Support

When you boot into the Recovery volume you have to log in with your user name and password. The computer knows you have admin privileges and lets you proceed.

To confirm that try logging in the Recovery volume with the credentials of a non-admin user as see where that gets you.

Yale Linguist wrote:

My wife's computer (a recent Mac Mini, still running Mojave) has gotten into an impossible state. When she (or anyone else) logs in, a dialog box appears with the above text, and "Enter your password to allow this". When you enter the user's password, and click "repair", it just puts up the dialog box again. Forever. And you can't do anything else. I have looked at other instances of this problem (mostly on High Sierra systems), but the suggested remedies don't work. In particular, the Apple-suggested remedy of opening Get Info on the home directory, unlocking Sharing & Permissions and then clicking "Apply to enclosed items", letting that run, and then restarting does no good (I've tried it three times with the same non-result). Apple then suggests that if this fails, reinstall macOS. Problem is, she's allowed so much stuff to accumulate on the machine that there's no disk space left, so "reinstall macOS" can't go through.

Any suggestions would be MUCH appreciated.

What Library is this in reference to (?)

Is this the exact message box ...

If this is your user Library (~/Library) then from your statement above:

"When she (or anyone else) logs in,"

this means every users ~/Library needs repair??? This is hard to understand.

https://apple.stackexchange.com/questions/341524/macos-needs-to-repair-your-library-error-loop-on-mojave-upgrade

I would try a safeboot— to see if you can log in from that environment to do some house cleaning.

Try a SafeBoot https://support.apple.com/en-us/HT201262

Takes noticeable longer to get to the login screen, does a 5-15 minute disk repair before it fully boots up, and certain system caches get cleared and rebuilt, including dynamic loader cache, etc. Login and test.

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I understand your point about preserving the unstable state of the Mac.

Please tell me what model year mini is that? And what version macOS?

I recommend you take a quick trip to OfficeMax or Staples and pick up an inexpensive external drive that you can use to offload a good chunk of her stuff by directly dragging, then delete the file from the Macs drive and thus free up that space..

I would not start copying stuff to the TM drive. That would imperil the backups there.

Meanwhile, you might try to download and run Malwarebytes to see if it recognizes any sort of malware on the machine. It's a small app, safe and free - you don't need the paid version for this.

Also, please consider as a troubleshooting measure downloading EtreCheck from the Mac App Store. Again, another free and safe utility. It gets a lot of use in the Community. EtreCheck gathers important debugging information about your Macintosh and presents it in a report that you can include in your reply here. It is very highly regarded and meant to be used with Apple Support Communities to help others help you troubleshoot and resolve your Mac problems.

Please navigate to the Mac App Store OR to EtreCheck.com and download the free version of EtreCheck. Once you’ve installed the app and created the report please post it with your reply to this message. 

For instructions on how to attach your EtreCheck report to your reply, please click  How to use Add Text when posting…

We can then use the EtreCheck report to look for the more obvious things that may be causing your problem and advise how to correct them.

Glad to hear you got this resolved. Well done! 🙂