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Let $F$ be a nonarchimedean ordered field. Is there always a hyperreal field $^*\mathbb{R}$ such that there is an embedding of $F$ in $^*\mathbb{R}$?
As far as I understand it, the answer here implies that the answer is positive in the special case where $F$ is the Dehn field or the Levi-Civita field.
A definition:* A hyperreal field is an extension $^*\mathbb{R}$ of $\mathbb{R}$ equipped with an operator $*$ defined on the structure $\mathcal{S}(\mathbb{R}):=\bigcup \limits_{n \in \mathbb{N}} \mathcal{S}_n(\mathbb{R})$ where for any set $E$, we have $\mathcal{S}_0(E)=E$ and $\mathcal{S}_{n+1}(E)= \mathcal{S}_n(E)\cup\mathcal{P}(\mathcal{S}_n(E))$ for all $n \in \mathbb{N}$. This operator fixes $\mathbb{R}$ pointwise and assigns a an element $^*X \in \mathcal{S}(^*\mathbb{R})$ to each element $X$ of $\mathcal{S}(\mathbb{R})$ (with $^*\mathbb{R}$ being the value of this operator at $\mathbb{R}$), such that the following transfer principle is satisfied:
For all (set-theoretic, first order) formulas $\varphi[x_1,...,x_n]$ where each quantifier appears in a bounded form $\forall x \in y,[...]$ and $\exists x \in y,[...]$, we have that for all $X_1,...,X_n \in \mathcal{S}(\mathbb{R})$, $\varphi[X_1,...,X_n]$ holds if and only if $\varphi[^*X_1,...,^*X_n]$ holds.
(in particular, $*$ preserves the "type" of sets, e.g. sends $n$-uples of reals to $n$-uples of hyperreals and so on)
One way (in fact, the way) to obtain this type of field is through ultrapowers of $\mathbb{R}$. One can also characterize them as quotients of rings $\mathbb{R}^I$ of functions $I \rightarrow \mathbb{R}$ by maximal ideals, where $I$ is a non-empty set. By a result of Keisler-Kunen, for all cardinals $\kappa$, there exists a ultrafilter $\mathcal{U}$ on $\kappa$ such that the corresponding ultrapower $\mathbb{R}^{\mathcal{U}}$ is $\kappa^+$-saturated. This implies that any ordered field of cardinality $\leq \kappa$ embeds into $\mathbb{R}^{\mathcal{U}}$. So the answer is yes in this sense.
See A ring of homomorphisms is enough to get non-standard analysis for more details. See also A purely algebraic characterization of fields of the hyperreal numbers for a somewhat more explicit description of those fields.
*this is different from Dale and Woodin's definition of hyper-real fields in their book Super-real fields. However the result is still true with this definition, since hyperreal fields in the previous sense are hyper-real fields in their sense.
