The Galaxy client doesn't show the 'More' button, that is a well known oversight in the client.

Use a browser like Firefox to log on to GOG on their webpage. Then hover with the cursor over your username on the menu at the top and click on 'Games'. This will bring you to your game library where you can download offline installers, extras, etc. with your web browser.

Find the game in your library and click on its game card. There you will find the 'More' button. Click on it and select 'Serial Keys'. See attached Screenshot.

Thanks, although I've already solved the issue without needing the key: running the Verify/Repair command in the Galaxy client got rid of the serial key promt.

Good to hear that you got it running.

However, Galaxy might have only inserted a generic key that is valid for singleplayer and you might need to use your unique key for multiplayer if you ever want to play against other people. That's why I always check if my newly bought games have a key and write them down to make sure I have all my keys in a safe place.