Is there a real category of question science cannot answer, or is that just not yet?

The clean way to cut this is by what kind of claim is being made. Empirical claims are about how the world in fact is, and those are science's business even when they are currently out of reach. Definitional and mathematical claims are settled by proof or by convention, so asking science whether a number is prime is a category error rather than a hard problem. Normative claims are about what ought to be the case, and no measurement resolves them: whether a painting is good, whether you should keep a promise that has become inconvenient. Science can tell you everything about what happens if you break the promise and nothing about whether you should, and that gap is structural rather than temporary.

Empirical, definitional, normative. Only the third is structurally outside. Everything else is instruments and time.

I would defend your side harder than you are defending it. The graveyard of things declared permanently unknowable is very large: Comte famously argued we could never know what stars are made of, and spectroscopy arrived within a few decades and told us exactly. The composition of the seafloor, the age of the earth, and the mechanism of inheritance were all confidently placed outside reach in their time. So the base rate for it is not yet is high, and the burden should sit on the person claiming a permanent boundary. I accept the normative category as genuinely different, but almost every other example I have been handed has been a not yet in a costume.

I spent an embarrassing amount of a dinner party arguing that consciousness was proof of a permanent limit, and got taken apart by someone who pointed out I was arguing from how confusing I personally found it. That is not evidence about the world, it is evidence about me. I now try to notice when I am promoting my own bafflement to a law of nature.

A useful practical test: ask what observation would change your mind. If your friend cannot name one, even in principle, even with unlimited money and a thousand years, the question is not so much unanswerable as not the kind of thing an answer would attach to. That is not a criticism of the question, plenty of questions worth asking are like that, it just tells you which tool you are reaching for.