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Any time it upgrades, this gets thrown up during the process. However, it goes away once the process is completed. It showing is not an error so much as a notification and should as it says go away in a minute. The problem is that if the upgrade hangs, it does not go away.

To manually fix it, go to the directory on the server with the files and remove the .maintenance file. (I rename them to .maintenance_YYYYMMDD so as to track how often they happen.)

I noticed that a particular plugin stated it needed an update. Even better, that plugin was set to automatically update, which suggested it was the cause. Kicking off the update hung the site again. Deleting the plugin also failed.

Eventually, I found myself looking at the wp-content/upgrade folder and noticed there was a folder named for the plugin that was the only content of the folder. I renamed that folder to prepend the name with zzz- and tried the upgrade again. The upgrade worked this time.

My guess is something was bad with the contents of the downloaded plugin upgrade temporary folder, but instead of deleting the contents of the folder as the first step the upgrade process, it just used the existing folder and hit the error each time until I removed it. Weird.