There are two simple constructions for creating arbitrarily large non-Archimedean ordered field extensions of the reals.
First given such a field one may consider rational functions over that field with $f(x)$ declared positive if $f(x)>0$
for all sufficiently large $x$. Secondly one can Dedekind complete the field by filling in all good/narrow cuts in the
field as detailed in
this discussion.
I would like to know if iterating these two constructions over the ordinals produces the surreal numbers.
Specifically we define $F_0$ to be the real numbers. For an ordinal $\lambda$ with predecessor $\lambda-1$, define $F_\lambda$ to be the
ordered field obtained by Dedekind completing the rational functions over $F_{\lambda-1}$. Otherwise if $\lambda$ is a limit ordinal,
define $F_\lambda$ to be the union of all $F_\kappa$ with $\kappa<\lambda$.
Best Answer
No. Think about the fields as valued fields under the natural valuation induced by the order (i.e., the valuation ring consist of elements bounded by an integer). Order completion then coincides (for nonarchimedean fields) with their completion as valued fields, in particular, it preserves the value group. If $F$ has value group $\Gamma$, then $F(X)$ ordered with $X>F$ as you are doing has value group $\mathbb Z\times\Gamma$ ordered lexicographically, in particular, it contains $\Gamma$ as a convex subgroup. Thus, value groups of all $F_\alpha$ will be discretely ordered. This means none of the fields nor their union can be a real-closed field, as real-closed fields have divisible value groups. (Specifically, the $X$ from $F_1=\widehat{\mathbb R(X)}$ will have no square root in any of the fields.)