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This is a line of thought about an algebraic approach to infinity I've been on and off since I was young. This is the closest I've come to deriving this "number" (?) $\omega$ in a logical way (I believe it's a different object to Cantor's $\omega$ but it certainly has some similarities.) Although I'm not happy about its completeness I'd like to share and make sure the bare bones are in the right place.

Let $\mathbb{Y}'$ be an arbitrary set of rational polynomials $\{f,g,h,\dots\}$ , and define some $\omega$ such that $f(\omega+k)\neq g(\omega+k)\neq h(\omega+k)\neq \dots$ for all $k > 0$.

For any two such functions, $f(x) - g(x) = 0$ at infinitely many points if and only if $f = g$, and so if $f \neq g$ there must be some largest $x$ where this happens. Hence, there must exist some $\omega$ for any $\mathbb{Y}'$ we choose. We end up with a well-ordered hierarchy of rational polynomials $$f(\omega)>g(\omega) \Leftrightarrow \exists\;\omega:f(\omega+k)>g(\omega+k)\;\forall\;k>0$$

The bit I'm unsure about is that $\omega$ is not necessarily finite: how to prove that this applies to the infinite entirety of $\mathbb{Y}$ if the $\omega$ we need is no longer a real number?

My question is simply the soundness of this argument: is the set of all rational polynomials well-ordered in this sense, what other sorts of functions can be included in the hierarchy, is it reasonable to use these abstract objects of the form $f(\omega)$ as nonequivalent infinite quantities (since $\mathbb{Q}\subset\mathbb{Y}$), and what gaps do I need to fill if I am right?

Supware
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I don't think what you're doing has anything to do with hyperreal numbers, the first infinite ordinal number $\omega$, or with well orderings.

However, I think you are approximating an idea that makes sense.

Consider the set of all rational functions with rational coefficients. These are the functions of the form $\frac{p}{q}$ where $p$ and $q$ are both polynomials with rational coefficients.

(we can use real coefficients too, or lots of other choices; the thing that matters most is that we have an ordering on the coefficient set)

(we could do this with polynomial functions rather than rational functions as well)

Rational functions form a reasonable algebra; you can add, subtract, multiply, and divide them and the results will still be rational functions. These operations will satisfy most of the familiar identities; in particular, they form a field. (don't divide by the zero function, of course!)

We can additionally define an ordering relation on rational functions in basically the way you are trying to describe:

$f < g$ if and only if there exists an $N$ such that $f(t) < g(t)$ for all $t > N$

This a total ordering, and it is compatable with arithmetic; with this choice of ordering, the rational functions form a ordered field. This example (but with real coefficients) is listed in the current version of that wikipedia page.

The rational function $w$ given by $w(t) = t$ is an example of a positive infinite number in this field. We say it is infinite, because for every ordinary real number $a$, we have a corresponding constant function $c_a$ given by $c_a(t) = a$, and this satisfies $c_a < w$.

(Normally we just write $a$ for the function $c_a$. A letter like $x$ is more common for what I called $w$ above)

You can extend this to a real closed field by adding in new numbers corresponding to finding roots of suitable polynomials.

However, all of this is only suitable for doing algebra. If we want to make sense of things like $\sin(w)$, we have to do a lot more work. At some point, it becomes much easier to just use the hyperreals.