Hi,
I downloaded the game yesterday and couldnt start it on my Windows 10 box with NVidia 8800 GT.
So I updated drivers, installed some suggested c++ runtime, restarted, didnt work.
Today I redownloaded the game, nothing.
Switched language of the game in the hope it downloads some different files which might work, nothing.
Closed GOG and renamed Galaxy.dll, nothing.
Afterwards I tried to run it as administrator, nothing.
In compatibility mode for Windows 8 and as Administrator, nothing.

It always crashes at launch, not leaving a crash report.

Do you have any hint on how to proceed?

I just googled your graphic card and it seems like it only supports DirectX 10. You need a graphic card which supports DirectX 11 to play Witcher 3.
"GeForce 8800 GT
Your search for the perfect graphics card is over — SLI™-Ready NVIDIA® GeForce® 8800 GT has arrived. The ideal combination of power, performance, and price, GeForce 8800 GT offers GeForce 8 Series goodness, 112 processor cores, and a 256-bit frame buffer for jealousy-inducing DirectX 10 gameplay at ultra fast frame rates."
http://www.geforce.com/hardware/desktop-gpus/geforce-8800-gt
But before buying a new graphic crad you should write down all your PC specs first.

Witcher 3
Minimum system requirements - Windows:
OS: 64-bit Windows 7 or 64-bit Windows 8 (8.1)
Processor: Intel CPU Core i5-2500K 3.3 GHz or AMD CPU Phenom II X4 940
Memory: RAM 6 GB
Graphics: Nvidia GPU GeForce GTX 660 / AMD GPU Radeon HD 7870
Please mind that we only officially support full-size desktop graphics cards
Hard Drive: 35 GB of available space
DirectX: 11

oh wow, those things age pretty fast, thanks for your time