Every time I close my browser (Firefox) I get logged out because next time I come back to the site I have to go through that Captcha to log in again.
This just started in the past couple of days. I never had this problem before, and it's not due to a Firefox update either.
I did come up with a work-around for this though. I installed Opera. It seems to let GOG remember me.
It's just a pain in the butt to have to install an additional browser just to use a website that worked fine until the recent website update.
It's really a shame too. I liked GOG until this website update.
I don't have that problem with the updated Firefox.
Check the privacy settings in Firefox and delete the GOG cookies.