In the entire time everyone has been complaining about this new captcha thing, I have been using both Firefox, Chrome and Galaxy to log into GOG every single day on and off all day long, restarting all 3 applications and my entire computer several times. I have yet to see a single captcha from GOG whatsoever.
The only thing I can think of is that the people who are experiencing this and are annoyed by it are deleting their cookies and/or HTML5 localStorage upon browser exit or session end, or they're using the "Private Browsing" or "Incognito" mode of their browser which also deletes cookies when the session ends. I'm not about to test that theory out myself because I'm able to log in to GOG in an ongoing basis right now as I always have without any captcha, but if someone is experiencing this issue I would recommend whitelisting GOG.com and its subdomains to allow cookies and localStorage permanently rather than session based or temporary.
If someone is worried about "being tracked", there are a multitude of ways that modern websites use to track people if they really want to and some of these techniques can not be filtered out with browser addons or refusing cookies etc. anyway (such as image metadata tagging and caching). Any 3rd party tracking/etc. tech can be blocked easily by various browser addons, and cookies are necessary for the site to work. If one is logged into their account, all of the data is present while logged in to be able to track someone between sessions if desired anyway regardless of cookies, so there is really no benefit to blocking cookies on the gog.com domain. It's safe to block 3rd party cookies of course, and that wont interfere with login, but I don't believe there are any anyway.
If someone wants to protect themselves from being tracked reasonably without causing websites to be broken, a combination of Request Policy Continued, NoScript, Privacy Badger, Cookie Monster, and Disconnect can be used with Firefox, or the same or similar addons with Chrome. All of the gog.com domain should be permitted through all of these addons which require explicit permissions be granted, and then things just work. One can still have more highly strict settings on other websites if desired and customize on a site by site basis.
Just a suggestion that I hope some may find useful, who may not have known about such options or used them before. Others may or may not be aware of these options and may not be comfortable or happy to do things this way of course, and that's fine. Enjoy captcha I guess. :)