It wasn't the cache.
But I believe I have narrowed it down to Chrome, which is the browser I use 99% of the time for everyday work. All the stuff below is from the Chrome Dev console and relates to one site I was using for testing which would not backup in Chrome. After it failing, and i went through the process described below, I loaded the site in Safari and ran Akeeba. Perfect. Then in IE 11, and ran Akeeba. Perfect. Just to be certain, I then did the same in FF 52. Again, perfect. In between backups on the non-Chrome browsers, I attempted a backup in Chrome. Failed each time. (In other words, I went Chrome, Safari, Chrome, IE, Chrome, FF, Chrome)
I repeated the above on another site which refused to backup in Chrome. A Gain, it backed up fine in the other browsers. However, I then did a 3rd site and it immediately backed up fine in Chrome. So I really don't know. At least I know I can get backups done using a different browser. Perhaps it's time to drop Chrome. I have noticed that, in the last 2-3 weeks, it's cache has become EXTREMELY sticky. Previously, F5 was generally enough, sometimes having to use Ctrl-F5. Now, I'd guess 1/2 the time I have to go and physically clear the cache to display changes.
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On one of the offending sites, with the console open, just navigating to the Akeeba Backup (without starting anything) page shows this error:
JQMIGRATE: Migrate is installed, version 1.4.1
DevTools failed to parse SourceMap: http://www.ckdev.info/mebizwiz/media/com_akeeba/js/UserInterfaceCommon.min.map
DevTools failed to parse SourceMap: http://www.ckdev.info/mebizwiz/media/com_akeeba/js/Modal.min.map
DevTools failed to parse SourceMap: http://www.ckdev.info/mebizwiz/media/com_akeeba/js/System.min.map
DevTools failed to parse SourceMap: http://www.ckdev.info/mebizwiz/media/com_akeeba/js/Tooltip.min.map
DevTools failed to parse SourceMap: http://www.ckdev.info/mebizwiz/media/com_akeeba/js/piecon.min.map
DevTools failed to parse SourceMap: http://www.ckdev.info/mebizwiz/media/com_akeeba/js/ControlPanel.min.map
DevTools failed to parse SourceMap: http://www.ckdev.info/mebizwiz/media/com_akeeba/js/Ajax.min.map
I started a backup on one site that wasn't working and got this from the console:
Starting backup Backup.min.js:7
Uncaught TypeError: Cannot read property 'replace' of undefined at Backup.min.js:7 Backup.min.js:7
Uncaught TypeError: Cannot read property 'replace' of undefined at Backup.min.js:7 Backup.min.js:14
Running backup step Backup.min.js:14
Object Backup.min.js:16
Starting tick timer with waitTime = 0 msec Backup.min.js:6
Timer tick Backup.min.js:8
Uncaught TypeError: Cannot read property 'replace' of undefined
at Object.akeeba.Backup.resetTimeoutBar (Backup.min.js:8)
at Object.akeeba.Backup.timerTick (Backup.min.js:6)
at <anonymous>:1:15
It stalled. Reinstalled AKeeba - no change. Exited Joomla and cleared cache - no change.
Interestingly, the sites which would not backup yesterday are now doing so ON CHROME.