Very interesting question; there may be some way. Depending on the scale of your application, you may consider investing the time in a proper setup. For instance, if the document is very large, you may consider splitting it into multiple small files. The “date modified” timestamp on each would then provide you with the info you need at a glance, in Finder or Windows Explorer. Then you could create three “full files” linked to all the little chapter files. (I’m getting this idea from the InDesign book writing workflow, wherein each chapter is a single file and the book is another file that simply includes all the chapters.)
I don’t know that there is any way to do this version tracking from WITHIN LibreOffice—I don’t think version control is LO’s strong suit. But the above method would be a decent workaround, depending on how long the document is and how unwieldy it would be to split it up.