Suppose, given $\epsilon \in \operatorname{monad}(0)$ and $\epsilon \neq 0$, is it true for each $x \in \operatorname{monad}(0)$, $$x = \sum_{r_i \in s \subset \Bbb R, s \text{ is finite}}a_i \epsilon^{r_i}$$ $a_i \in \Bbb{R}$ is a constant coefficient.
Let $S$ be a subset of $\Bbb {R^{\Bbb R}}$ such that for all $f \in S$, $f(x)= 0$ for all but finite elements in $\Bbb R$
If so, it seems to me $(\operatorname{monad}(0), \leq)$ is isomorphic to $S$ with a lexicographic order.
Suppose that $\epsilon$ is a positive infinitesimal. In the ultrapower construction let $\langle x_k:k\in\omega\rangle$ represent $\epsilon$; I’ll denote this relationship by $\epsilon=\langle x_k:k\in\omega\rangle_{\mathscr{U}}$. Without loss of generality we may assume that $x_k>0$ for all $k\in\omega$. Then $\epsilon^n=\langle x_k^n:k\in\omega\rangle_{\mathscr{U}}$ for each $n\in\omega$. For $n\in\omega$ let
$$y_n=\frac12\min\left\{x_k^n:k\le n\right\}\;;$$
then for each $n\in\omega$ we have $y_k<x_k^n$ for all $k\ge n$, so if $\delta=\langle y_k:k\in\omega\rangle_{\mathscr{U}}$, then $\delta<\epsilon^n$ for all $n\in\omega$. Thus, $\delta$ is smaller than any finite real combination of real powers of $\epsilon$: it’s infinitesimal compared with $\epsilon$.