A free tool called DiskInternals ZIP Repair was able to recover a corrupted PowerPoint file (*.pptx) that I had on my computer.  I learned that pptx files are actually zip files in disguise, and so using a utility to repair a corrupt zip archive could work.  At first I thought it was one of those sketchy bloatware programs, but Lifehacker has written an article about them before, so it should be fine.

Using the program was pretty simple.  I just opened it and told it which zip file (in my case the pptx file which I renamed to a zip file), and then the program did all the work and recovered everything.  It was like magic!