macOS Yosemite->High Sierra update not sticking I need to download exam software for my computer, but the software does not support Yosemite, my current OS.  Preposterously, this $100 software costs $50 to "redownload", so I need to be absolutely sure it will work on this computer before trying it out.

I went to the App Store, checked "Updates", confirmed that macOS High Sierra was in the roster, and clicked the button to commence the update.  My computer shut down, I saw the expected loading bar, my computer restarted, everything looks fine.  

I check "About This Mac" and it still says OS X Yosemite 10.10.5.  www.whatsmyos.com says I still have OS X 10.10.5. 
Typing "sw_vers" into the terminal prints out "ProductVersion:  10.10.5".

I check the App Store, and under "Updates Installed in the Last 30 Days" there is a row that clearly reads "macOS High Sierra" and "Installed May 6, 2019".  The top of the page reads "No Updates are Available" so I can't attempt to update the OS again.

What's going on here?
"What's going on here?," Indeed.

I can't guess, it could be a damaged disk or disk structure, permissions issue, yadda yadda yadda...

But there are a few things you can try. Note these suggestions are going to be kind of generic due to the nature of the problem and because you didn't include your Mac model in the question. It would be best if you added that to your original question.

Boot into Recovery Mode ⌘ CommandR when restarting your Mac. When you have a menu bar available start disk utility, select your drive and click on First Aid. If the drive shows clean when First Aid has completed you can quit and reboot normally. If it shows problems (fixed) then re-run First Aid until you get no errors. If it is unable to fix, add the error messages to your original question.

Now, look in your Applications folder for an app called "Install Mac OS High Sierra" or something like that. Once you have verified you have the app I would use the freeware installer creator DiskMaker X a blank 8GB USB key and make a bootable High Sierra install key.

Once you have that, insert it in any USB port and reboot your Mac holding the ⌥ Option key. Select the installer USB key when the boot disk selector appears and let the Mac boot from that. It will take a little longer than usual.

Follow the prompts to install High Sierra. Important to note you should NOT use Disk Utility to reformat/repartition your drive, just install right over top of what is already there. An upgrade install.

Report back on those results.

Note if that fails you may need to back up your Mac and use the installer to reformat the boot drive and then install High Sierra and restore from your backup.