Sorry for the necro, but I feel like it's worth mentioning:

NoScript detects 4 script domains at gog.com/redeem:

gog.com
gog-statics.com
googletagmanager.com
recaptcha.net

I blocked recaptcha.net while allowing the other 3 domains, and when I tried to redeem a code, I was not locked out by recaptcha. So either recaptcha is no longer mandatory, or recaptcha is tied to another domain or script.

If someone could confirm it, it'd be great.

Since this thread has been resurrected, I feel I should point out that, each time a reCAPTCHA fails to correctly accept a human user, the user should e-mail support@gog.com about the issue. They will try to claim it's your fault the first couple of exchanges. Keep at it. Be polite, but firm. Remind them that their broken implementation is impeding a sale. If you keep at it long enough, they might take an interest and help you. At the very least, you need to make them understand how much trouble their terrible implementation causes.

It must be something to do with settings, as I am struggling to remember the last time I got presented with CAPTCHA at GOG. I don't see it during Redeem etc. I seem to recall the last time was when I rushed, before the web page had finished loading, so I take my time now.

On the gift code redemption page, I've never been able to use the "Continue" button to view what a code provides. The button starts out disabled and, until the most recent redesign, the page was inexplicably broken. With the recent change to cite Google's terms of service, it's now clear to the user that a captcha is meant to be in the page. It still doesn't work though, because it's hosted on a blocked domain. Even if I get the domain unblocked (recursively, through several levels, none of which are properly documented in the page), it still doesn't work, because it's completely unsolvable. No matter how many I solve, it's never accepted. Ever. Due to the completely user hostile nature of the gift code page, I neither accept gift codes nor buy them for anyone else, because I can't trust that my intended recipient would be able to redeem the code, and if they can't, then buying the code is a waste of money.

I also cannot use the web support portal due to a captcha.

I do not recall encountering a captcha on login, but I've been logged in continuously for a long time.