That my point. I don't want to re-log constantly. I'm the only one using GOG on my computer and I want to be logged in always like before rather than re-entering my password every time I visit the site.

As mentioned here

Did you try? For some people it helped.

I do remember that one other user had problems with one of the addons. Did you try disabling them all?
For this user it help and he was even able to reactivate all addons afterwards again?

You might want to try disabling the addons first before deleting all cookies ;)

Strangely, switching from https to http seems to have helped. It's working so far, I hope it remains like this.

Thanks for all the help, guys!

Clear the GOG cookies and the browser cache, restart the browser, then log back in. If you are using a plugin like NoScript, disable it completely until after you login, then re-enable it and whitelist GOG. In most cases, this should correct the problem permanently... or at least until GOG breaks the login again.