Can my Mac mini update beyond macOS 10.8.5?

How to download and install macOS - Apple Support

If your Mac mini is compatible with High Sierra or Mojave or Catalina, and you want to install one of those three, I believe that you will first need to upgrade it to El Capitan or Sierra. The versions of Safari and of the App Store in older versions of Mac OS X might not be able to access the Support document or use the App Store links in it.

Even just to get from Mountain Lion to El Capitan or Sierra might require downloading the .DMG installer, using a modern Mac or PC, then transferring that file (on a USB drive) to the old Mini, to actually open it (mount the disk image) and run the installer program within.

Whether and how far you can upgrade your Mac mini will depend on its hardware model year.

If the hardware model year is

• 2009 – You can upgrade it as far as El Capitan (10.11.*).
• 2010 or 2011 – You can upgrade it as far as High Sierra (10.13.*).
• 2012 – You can upgrade it as far as Catalina (10.15.*). Catalina will not run 32-bit applications, and fell out of "the most recent three" some time ago, but is recent enough to run current versions of Firefox and LibreOffice.

If the hardware model year was before 2009 or after 2012, it wouldn't be able to run 10.8.5 (Mountain Lion) now.

It can be upgraded at least as far as El Capitan; you may need to use a different computer to download the installer.

(260057)