If you have an Office Open XML document with the extension .docx (or .xlsx, etc.):
The file is a .zip archive, so open with TotalCommander, or WindowsExplorer (copy first as a .zip), or 7zip. Open file 'docProps\app.xml' in the archive and look for
<AppVersion>12.0000</AppVersion> is Office 2007
<AppVersion>14.0000</AppVersion> is Office 2010
<AppVersion>15.0000</AppVersion> is Office 2013
<AppVersion>16.0000</AppVersion> is Office 2016
Found out by saving .docx from Word 2007, Word 2010 and comparing. Later found this info on the web too. For easier googling let me refer to that comment too:
comment from Daniel Klein on 14 Aug 2012, 19:45
Later versions of Office documents (after Office 2003) are saved as ordinary zip files but with a different extension (e.g. ".docx"). You can view the contents by first making a copy of the file and changing the file extension to ".zip". It will then open in any zip viewer.
The relevant information is in ".\docProps\app.xml". Extract this file and open it in a text viewer. At the end of the file is something like
<AppVersion>14.0000</AppVersion></Properties> which tells you the file was last saved with (in this case) Office v.14 (Office 2010).