Hyperreal field extension

There isn't any trouble constructing $\mathbb{R}(\epsilon)$ as a formally real field that is isomorphic (as a field) to $\mathbb{R}(x)$, and with $\epsilon$ a positive infinitesimal. (it may be easier to see the ordering by writing a rational function as a formal Laurent series in $\epsilon$ about 0)

It's easy to see that this is not a hyperreal field: it only has finite powers of $\epsilon$. In particular, if there were a positive transfinite positive integer $H$, then there would be an element $\epsilon^H$ that is smaller than every power of $\epsilon$ appearing in $\mathbb{R}(\epsilon)$.

($\mathbb{R}(\epsilon)$ also fails to have a square root of $\epsilon$. This can be fixed by passing to its real closure, but that would still fail the above property)

If you tried adjoining a second infinitesimal, a similar argument proves it can't be hyperreal.

Briefly, one way to approach this question would be to try to construct alternative models of the hyperreals via the Compactness Theorem that either satisfy or fail to satisfy the properties you want. The Enderton text, A Mathematical Introduction to Logic, uses this kind of construction and has a very rigorous treatment of nonstandard analysis that you can use as a guide.

Since I only have 44 minutes left on this question, it could easily contain errors so I'm retagging your question to include logic and set theory so it can be properly checked.