On the other hand, it is interesting to think about the fact that, say, the Avogadro number is usually represented as a real number; the problem with the number of protons or other particles is that their number is unbounded: given enough energy, another particle can always jump into existence...

That's still arbitrary (designed by humans) because you decided to count each proton as 1. We could also count every proton as 2 and it would be just as arbitrary.

Unfortunately, the number of dollar bills in your wallet cannot be thought of as arbitrary.