Actually, it was high school geometry where I found my calling as a mathematician. It was the first course in any field that I found actually interesting. So, it was good for inspiration at least, if not for calculus.

@DanielR.Collins, the discussion I give of the flipped classroom is, however, based on about 20 years of teaching experience.

@MichaelDNguyen, I still think the answer applies. In US education, at least, an undergraduate education is very broad and only begins specialization. Other places it is more specialized to a single field. Keep options open.

@Number, I'm not disputing that. The question tag is secondary-education. I was trying to frame something that would be easily understandable at that level without going deeply into the details. I thought I put sufficient caveats into my answer. I didn't intend, nor would I, at this level try to teach algebraic number theory to HS students when teaching them about unique factorization. It is enough, IMO, to show that there are systems in which it doesn't work. I interpreted the question of the OP to be seeking that, just as much as the explicit question. Read my final sentence in the answer.

@Number, yes, and so I put "prime" in quotes. Not-composite would be better, of course. I'm not sure I agree with you about introducing the integers mod 6, though. You don't need to give a complete treatise on them, or even introduce the terms Group or Ring. But the complete multiplication table is trivial to show so the example of something "similar" to the integers that doesn't have a fundamental theorem is pretty easy to demonstrate. This can show the importance of it in the Integers.