I want to find a construction of ${}^*\mathbb N$, that does not involve the reals, because my interest is in using the hypernaturals as a step towards an alternative construction of $\mathbb R$ as a subset of ${}^*\mathbb Q$. My reasoning is that since ${}^*\mathbb N$ is elementarily equivalent to $\mathbb N$, the following construction should be more intuitive and believable then a construction that starts by "observing reals are equivalent to their decimal expansions" or any similar assumption. I believe the same could be said of ${}^*\mathbb Z$ if this is easier.
For context this isn't a nonstandard analysis question (which seems to have developed its own standard) and I'm hoping that the construction can be described mostly in layman's terms, and preferably with a few examples that explain which numbers are and are not considered part of ${}^*\mathbb N$. Perhaps in the context of ordinals.
My thought process so far has led me to some idea that ${}^*\mathbb N$ can be constructed as an ordered subset of $\mathbb N^\mathbb N$ (ordered infinite sequences of naturals equivalent to functions over $\mathbb N$), however since I haven't been able to pin down exactly what elements are supposed to be ${}^*\mathbb N$, I haven't been able to convince myself that any of the subsets I've looked at even have the correct cardinality. for instance multiplicative, additive, and subtractive combinations of $c$, $x$, $x^2$, … $x^x$ etc. seems to still only be countable.
Identifying a particular subset of $\mathbb N^\mathbb N$, and defining operations on it, is going to be... tricky.
For example, assume that $x=(0,1,2,3,\ldots)$ is in your set. Is $x$ an even number or an odd number? If you define addition on $\mathbb N^\mathbb N$ as simple coordinate-wise addition, then it is neither; there is no $y$ such that $y+y=x$ or $y+y+1=x$. But then, $x$ isn't a hypernatural number!
You might try to rescue the situation by saying, we'll declare that $x$ is even, and $x/2=y=(0,0,1,1,\ldots)$, and to make that work, we'll identify $y+y=(0,0,2,2,\ldots)$ with $x$ on the grounds that the two sequences agree infinitely often. Ah, but they also disagree infinitely often... Well, more specifically, the two sequences agree on even coordinates, and we're arbitrarily choosing to declare that when comparing sequences, the even coordinates matter more than the odd coordinates. Notice that by identifying different sequences, we are no longer choosing a subset of $\mathbb N^\mathbb N$, but rather a quotient of $\mathbb N^\mathbb N$ by some equivalence relation.
Next someone will ask you whether $x$ is a power of $2$; whether it's a factorial; whether it's a Fibonacci number...
...and you'll have to do the same thing for every dichotomy of the natural numbers: to decide which side of the dichotomy matters more, and to do it in a consistent fashion, while ensuring that $x$ never becomes equivalent to a finite number...
...and long story short, you've wound up with a free ultrafilter on $\mathbb N$, and your hypernatural numbers will be the corresponding ultrapower. This is the standard construction. Further info: https://en.wikipedia.org/wiki/Ultraproduct