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I don't quite understand your question, but in case it's at all useful, a standard way to think about primes is that they're the atoms. It's not that they're left out of multiplication / division, but rather that they are irreducible. [1/N]
The tricky part is proving that every integer is uniquely (up to reordering) expressible as the product of primes. Then your question about power primes going on forever can be thought of as a question about special factorizations; [2/N]
that is, ones where all the exponents are the same (if n = a^b, then you can factor a into primes and distribute b over them to get the [unique!] prime factorization of n). I think it's best to leave the questions about obviousness to you; [3/N]
I've studied math long ago to have lost trust in my sense of what's obvious in questions like this. Sorry if I misunderstood your question! [4/4]
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