Do you have any Word addins? Try disabling them. If solved, re-add one at a time until you identify the culprit.

If that doesn't work, backup your Registry. The Recent lists are stored there, and you're going to change it.

Run Regedit and go to:
HKEY_CURRENT_USER\Software\Microsoft\Office\16.0\W ord\User MRU
[the "16.0" will be different for you, probably 13 or 14]
[the "User MRU" might be only "MRU"]

The above is for my Word in Office 365 on Win10—the main thing you're looking for is "MRU", which stands for Most Recently Used. You'll find it in the other Office apps registry entries too, and maybe at various other locations within the Registry.

When you find above, delete whatever's in it & close Regedit.
Open a couple of Word docs.
Close Word.
Do those now show up in Recent?

Word recreates the MRU entries and changes them as you work with docs. You can compare the Registry entries of Word & Excel, if you want to see what a working MRU looks like when it's at home.

Fyi if you get to Office 2016 or 365: Outlook now has MRU for its 'Attach file' function—no more hunting thru your folders to find a file you just closed. Nice time & frustration saver if you email many files per day.