I'm not sure about this part of your description:

maine15 said

For past 3 weeks or so if I type Amazon into browser and click on Amazon link I am prompted to type some crooked letters and numbers to "prove I'm not a robot." I usually just hit the back arrow and do it again and it works.

To get a fuller picture:

(1) Where is the link you're clicking to get to Amazon -- is it on a search engine results page?

(There is some old malware that had a pattern of diverting you to a fake or unrelated site on the first click but when you went back, allowing you to go to the real site on the second try, sometimes called the Google Redirect virus. But if it only affects one site, that kind of infection seems unlikely.)

Does the problem still occur if you follow a full link instead of using a search engine result: https://www.amazon.com/

(2) Is Firefox's Password Manager (or an add-on password manager) trying to log you in using a saved password as soon as you hit that first link?

(Amazon could possibly be detecting and treating as suspicious "robot-speed" logins, but I haven't searched to see whether that is a thing.)

I tried getting help from Amazon but they said this is not currently a issue with other people and I should try clearing cookies and using HTTP when accessing Amazon homepage!

It's happening to at least one other Firefox user:

When I signin to Amazon.com, it sends a new page where i need to reenter my password and a string of characters - unsolved

I go to Firefox home page and type "Amazon" I get many, many "hits" and I click on the second one (I always ignore the one that says "Ad" next to it.) I copied the second one:

Amazon.com: Online Shopping for Electronics, Apparel, Computers ... https://www.amazon.com

So I click on it and I get an Amazon page saying they need to be sure I'm not a robot and I should type the letters/numbers I see below. Rather than do that I hit the back arrow and then when I go back to Amazon I'm all set. No trouble accessing home page.

I've been ignoring this for the past 3 weeks but today I did some experiments. I tried it on other laptop in my home that I rarely use and it happened with the Firefox browser. I also tried both Explorer and Chrome. It did not happen with those browsers.

Since I've been using Amazon for at least 8 years and this is a relatively new problem, I thought I'd try to find out what the problem is. As I said, the 3 different people I "chatted" with at Amazon were not really able to help.

I did search for similar problems and found a few instances where people have faced a similar problem, but with different websites (such as Yahoo). I have no trouble with other websites, just Amazon. Which is why I started with them.

What happens if you visit the link in a Private window? You should be able to do that by right-clicking the link on the search engine results page and choosing Open Link in New Private Window. Private windows have their own separate cache and cookies, so are not influenced by data previously stored by Amazon on your Firefox.

Ok, I tried the private window option and had the same result. I also did an experiment and cleared history/cookies from my largely unused Chrome browser on my old laptop that I really never use for Amazon and had the same result -- told to "prove I'm not a robot." So, I guess this is not a problem unique to Firefox as I first thought. To clarify, I titled my question to say this problem occurs when I try to log-in to Amazon. It actually happens when I try to access Amazon home page. I've contacted Amazon again and they have decided to "create a ticket."

I got the CAPTCHA page yesterday when I was following a link from somewhere to a particular product on Amazon. I didn't stop to figure out what was unusual about the link and now that I look at my history, I have no idea which one it was...

In the interest of closing this discussion, I had an email last night from Amazon saying they had solved my issue and it appears they did! They provided no explanation but I no longer get a CAPTCHA when I try to access Amazon home page.

That's good to hear. Well done. Please flag your last post as Solved Problem so others will know.