Should I not compress my jpeg files before uploading to avoid double compression?

Re-compressing a JPEG using the exact same settings does not result in quality loss beyond the initial compression. However, if the settings used by WordPress differ from those applied by your photo editing software, progressive degradation of the image may occur.

The safest route to mitigate this potential degradation is indeed to upload your images at full quality.

You could set the default quality level to match what you use in your photo editing software via the jpeg_quality filter - however, minor variations in the compression settings can still result in degradation. If you choose to go this route you might verify fidelity by checking a hash of the file output from your editing suite against that of the same image uploaded to WordPress.

More on JPEG compression and degradation can be found over on the Photography Stack.

From looking at WordPress' sourcecode it appears to use the PHP function imagejpeg to save images. The PHP documentation doesn't mention anything about checking the quality before saving the image.

So I did a test using the identify tool in ImageMagick and uploading images to WordPress.

Image 1: 96% quality before upload & 82% after upload (as expected)

Image 2: 50% quality before upload & 50% after upload.

This admittedly naive test implies (to me at least) that image 2 was left untouched. If it had reduced the quality I'd expect something like 40% quality.

If you want to showcase high-quality images on your website, then you can turn off image compression in WordPress.

You can do that by adding this part of code in your functions.php :

add_filter('jpeg_quality', function($arg){return 100;});

Or you can create your plugin :

 <?php 
    /**
    * Plugin Name: Remove WordPress Image Compression
    * Plugin URI: [Plugun URI}
    * Version: 1.0
    * Author: [Your name]
    * Author URI: [Your URI]
    * Description: This will remove the compression WordPress, applies to images when uploading to your media library.
    */
    /**
    * Override the default image quality when resizing and cropping images
    */
    add_filter('jpeg_quality', function($arg){return 100;});

The code above sets the value to 100. It means that WordPress compresses the image at its highest quality.

Now, after activating the plugin, any new image uploaded will no longer be subject to WordPress image compression. Keep in mind that was previously uploaded to activating your plugin will need to be re-uploaded to have the compression removed.