Sure. Sadly, my google search history is turned off, so I don't have a list of my recent searches there.
One that I remember from a couple months ago was that I was trying to recall the name of an exercise they used to do (and may still) in design schools. In the exercise, students are given a problem and a short period of time to work on it, everybody working at the same time. At the end, they get together and critique each other's solutions.
My search terms in both Google and Kagi would have been something like "design exercise with simultaneous work and followup review".
If I run that search now, Kagi gives me articles about psychological studies that combine exercise with learning to test whether you retain more information. It interprets "exercise" as being physical exercise, which is the wrong sense of the word in this case. I think it interprets "design" as being the design of the study, which is also the wrong domain. It does give some articles about graphic design a little way down the page.
With the same terms, Google at least gives me articles about design related activities involving teams. At a glance (and in my memory of the original experiment) none of the articles give me the word I was looking for, but at least it knows the domain I'm searching in.
I remember this example because this was the first time I successfully used ChatGPT to ask about something, and it actually gave me the perfect response, with no fuss, and completely blew all the other systems out of the water. I gave it a natural language version of the same search as above, and it instantly told me the answer ("charrette") along with an accurate description.
So, maybe a bad example, because neither search engine was correct. But, Google definitely got me closer, and I suppose that if I did not have a very specific word in mind, it might have helped me by giving me examples of similar things. Kagi, frustratingly, seemed to have zero idea what I was asking about, and I could not even get it in the ballpark. I wish I could remember (or conjure up) a better example, but that's all I've got at the moment!