How would you feel about having a small box you can tick when you right click a game. It would add to the main view of all the games but place a ribbon going over the corner of the games image saying "Complete".

Another idea is for GOG to implement something you can click and have GOG randomally select you a game to play out o the games you own, and that you can get a recommendation based on catagory.

You can always just tag the game as "completed" or "played".

What I do is to simply rate completed games using the integrated 1-to-5 star rating. That way, if a game is unrated, it's because I haven't completed it yet.
That also lets you create dynamic collections in the left side of the client by shorting by "Unrated"

Yeah, I mostly used tags for that. I addition to "finished", I've also have tags for "to play again", "to mod", "to buy DLC"... all that could or not go along side the "finished" or "completed" tags. :)

Great idea..i have another frontend that has that feature,LaunchBox