I have discovered an older iMAC with OS X Version 10.9.5. Is there any hope of updating at least to Lion or Mountain Lion?

Back up your data, check your applications for compatibility, and depending on which iMac model you have, download El Capitan, High Sierra, or Catalina.

A Mac running 10.9.5 automatically meets the requirements for El Capitan. High Sierra requires the model identifier to be at least iMac10,1, and Catalina requires it to be at least iMac13,1.

Lion and Mountain Lion would be downgrades.

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Here is a Link to High Sierra 10.13:

How to upgrade to macOS High Sierra - Apple Support

The link is near the bottom of the article.

Get the Model Identifier by holding down the Option key and selecting System Information from the  menu.

Then we can help.

That depends on exactly what model iMac it is.

You have a 21.5" Mid or Late 2011 iMac. You can run High Sierra. Alan's post describes how.