Cannot defrag my computer since installing Windows 8.1?

Have a look at the earlier suggestions in this post.  Maybe the Safe Mode or Clean Boot is worth a try?

Are you running any third party disk monitoring types of programs?  Are you running the Windows "File History"?  I'm thinking that something like the Volume Shadow Copy service might be running at the time of Defrag, or something similar.

How to start Windows 8 in Safe Mode

http://www.bleepingcomputer.com/tutorials/start-windows-8-in-safe-mode/

How to perform a clean boot to troubleshoot a problem in Windows Vista, Windows 7, or Windows 8

http://support.microsoft.com/default.aspx/kb/929135

Earlier if the post there was talk on running Defrag from a Command Prompt in Windows.  It might be possible to run it from a Command Prompt outside of Windows via the Startup Repair options?

How To Access Advanced Startup Options in Windows 8

http://pcsupport.about.com/od/windows-8/a/open-advanced-startup-options-windows-8.htm

Understanding and accessing Windows 8's advanced boot options

http://www.geek.com/articles/chips/windows-8-advanced-boot-20121218/

You could try turning System Restore off on all drives and see if that makes any difference.

See Control Panel\All Control Panel Items\System\System Protection\Protection Settings.

...well, I did turn system restore off, run clean-up disk again, run defrag again and... the 3% fragmentation remains. When defrag runs, it appears like that something stops it from completing!

You could try turning System Restore off on all drives and see if that makes any difference.

See Control Panel\All Control Panel Items\System\System Protection\Protection Settings.

What about creating a System Restore point and then running the Disk Cleanup program and checking all the available boxes but more so checking the "More Options" Tab and removing all the old System Restore points except for the last created restore point.

After that see if Defrag will run.

Thank you - I followed the above and the results are:

 - Defragmentation analysis dropped down to 3% (from 16%). However, running defrag again, does not improve the score. I would have expected that it would drop down to 0%

What about creating a System Restore point and then running the Disk Cleanup program and checking all the available boxes but more so checking the "More Options" Tab and removing all the old System Restore points except for the last created restore point.

After that see if Defrag will run.