It mostly happens using "site:" queries which I use frequently to limit things to local websites (by domain) or for searching sites that have poor search engines (Amazon, for example). It rarely happens the first query, but often by the third or fourth modification or by the third or fourth page of results you visit, it will show a reCAPTCHA if it doesn't have enough "randomness" or doesn't think you're actually browsing Google and third-party sites the way others commonly do. (Robots are more likely to use search operators, for example, and more likely to pretend to be iPhones so they don't have to move the mouse, etc.)
My earlier query triggered it. Without a query, I can make the following text show up by going to https://www.google.com/sorry/index which when a relevant query is attached to the URL, it shows a reCAPTCHA for the search query, and also shows your IP address, etc.
> About this page
> Our systems have detected unusual traffic from your computer network. This page checks to see if it's really you sending the requests, and not a robot. Why did this happen?
If you click the link "Why did this happen?" it says:
> This page appears when Google automatically detects requests coming from your computer network which appear to be in violation of the Terms of Service[1]. The block will expire shortly after those requests stop. In the meantime, solving the above CAPTCHA will let you continue to use our services.
> This traffic may have been sent by malicious software, a browser plug-in, or a script that sends automated requests. If you share your network connection, ask your administrator for help — a different computer using the same IP address may be responsible. Learn more[2]
> Sometimes you may be asked to solve the CAPTCHA if you are using advanced terms that robots are known to use, or sending requests very quickly.
[1]: https://www.google.com/policies/terms/
[2]: https://support.google.com/websearch/answer/86640
The annoying part is that my account has never been whitelisted based on good behaviour. Instead, I end up seeing such reCAPTCHAs thousands of times a year, to the point where I stop counting them. Roughly half the time I'll answer the reCAPTCHA and the other half of the time, I'll close the tab and go do something else. Cloudflare site loading captchas are even worse, though. They delay the site by 5 seconds while they "check my browser", and then show an hCAPTCHA to solve, even when I'm already signed in with the first-party site. Very annoying, though the captcha is often easier to solve than Google's. The Cloudflare block often on streaming media websites. Ironically, Cloudflare's captchas have never prevented me from using commonly available Python scripts to watch streaming flash videos in VLC, they only block my web browsing...
I can only assume that Safari's excellent ad blocking and tracking prevention is causing my browsing traffic to stand out compared to others', enough that it prompts these CAPTCHAs more frequently.