Nvidia Shield Android TV works with GOG games. Ask me if your stuck on how to set it up.
I recently purchased one of these, and I can now stream my GOG games such as Pool of Radiance, Theme Hospital and Dungeon Keeper to my living room TV. My PC is located in a different room in my house aka my Computer Den. So now I stream my games over my WiFi Network, from my computer room, to my big screen TV in my living room. As a bonus I can stream my games from Steam too. If you were thinking of getting one of these and wondered if it works with GOG games, the answer is yes. I was on windows 7, then upgraded to windows 10, works under either one.

Feel free to ask me questions if your stuck or if you have specific questions about my set up.

I've actually been toying with one of these recently myself. I've a question though, as you brought it up... How much luck have you had using the gamepad with GOG games you've added to the Nvidia Experience application? I've tried a handful of games now, and most seem to open fine, but to be unresponsive to the gamepad. I got the impression GOG games that I could play with a bluetooth keyboard and mouse may work fine, but that gamepad games, overall, weren't going to work... perhaps something obvious I'm missing that you're aware of?

In college I used a projector to play Warsow.

The answer is a little complicated. I play mostly mouse/keyboard games from GOG using a USB wireless keyboard plugged into back of the NSATV. Those work fine.
As for the gamepad, I don't have any gamepad games on GOG that I currently play. I however have 2 gamepad games that I play from steam. ( Batman, arkham asylum ) and Mark of the Ninja. Both of those games work flawlessly when streaming from Nvidia Shield Android TV to my living room TV. I do recall once trying to play a third gamepad game and it did not detect I had a gamepad attached to my PC, so it defaulted to keyboard & mouse. Then an idea hit me.

Plan 1: I plugged my xbox 360 controller directly into the PC, so that the game detected that I had an actual gamepad. Meanwhile in my living room, I was using my Nvidia controller to play the game, with my xbox controller just remaining idle plugged into the PC just so that the games detection routine could detect a controller. I hope that tip helps you out.
Let me know if it does.
alternatively if you are still having difficulties

Plan 2:Try tricking it by adding the GOG game to steam , then create a shortcut from steam to the game on your desktop adding the steam shortcut to NVidia experience.

P.S. I am replying to this from a web browser I opened up thru NSATV after playing a GOG game.
It's awesome seeing a full sized browser on my big screen living room TV. I'm using a wireless keyboard & mouse USB plugged into the NSATV.

Thanks for the reply. I just tried your "plan 1" suggestion, though I tried it with a Wireless Xbox 360 for Windows controller. Turning the controller on, then trying to stream a couple gamepad-based GOG games I'd added through the Nvidia Experience application, but sadly, the Nvidia Shield gamepad still didn't work. However, my computer is apparently close enough to the television in the living room that I can just bring the wireless 360 controller into the living room, and *that* works once the game is streaming.
I have a wired 360 gamepad somewhere, so I'll try it with that once I can find it. But it's good to know that, if nothing else works, I *can* at least just grab the wireless gamepad from my PC and use that for GOG games, and the Shield gamepad for Steam games and for navigating the Shield unit itself.

I'll also try your "plan 2" suggestion today and try adding one of the games to Steam, just to see if opening it on the Shield through the Steam Big Picture mode will have any effect.

Thanks again for the suggestions! Though not ideal, even if nothing else works, knowing I can just grab the gamepad from my computer and still sit out in the living room to game sometimes is pretty awesome. :)

By the way, what browser are you using on the Shield TV?

Your welcome I'm glad i was able to help inspire you :) Just using your controller that is wireless plugged into your PC would have been plan # 3 . I was hold off to see if plan 1 or 2 worked for you before suggesting it.
I'm using google chrome. ( I also tried using Windows Edge that built into windows 10, but I prefer google chrome )

All I had to do was make a shortcut for google chrome on my desktop and then add it to my nvidia experience.
Let me know if your stuck on that and need more info.

P.S. feel free to friend me on steam