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A theorem due to N. Alling (Foundations of Analysis over Surreal Number Fields, §6.55) states that the surreal numbers are isomorphic, as an ordered and valued field, to the field of Hahn series with real coefficients and value group the surreal numbers themselves. There is also a restricted version, which I'll refer to in order to avoid the (IMHO uninteresting) foundational difficulties in dealing with classes: if $\kappa$ is a regular uncountable cardinal, the set $\mathrm{No}_\kappa$ of surreal numbers with birth date $<\kappa$ is isomorphic to the field of Hahn series of length $<\kappa$ with real coefficients and exponents in $\mathrm{No}_\kappa$ itself (in the indeterminate $\frac{1}{\omega}$).
Let us define $F_0 = \mathbb{R}$ and inductively $F_{\alpha+1}$ to be the field of Hahn series of length $<\kappa$ with real coefficients and exponents in $F_\alpha$ (the indeterminate being written $\frac{1}{\omega}$); and for $\delta$ a limit let $F_\delta = \bigcup_{\alpha<\delta} F_\alpha$ with the obvious embeddings. Then if I am not mistaken, $F_\kappa$ is indeed isomorphic to Hahn series of length $<\kappa$ over itself, it is indeed an $\eta_\xi$ field for $\kappa=\omega_\xi$, of cardinality $2^{<\kappa}$, just like $\mathrm{No}_\kappa$, and it is quite conceivable (I didn't check) that the two are isomorphic (as ordered—and valued—fields over $F_0 = \mathbb{R}$). But this can't possibly respect the map $x \mapsto \omega^x$ because in $\mathrm{No}_\kappa$ the latter has plenty of fixed points whereas in $F_\kappa$ it has none. So this construction is “wrong” in that it doesn't explain surreal numbers properly.
Thus, my question is: is there some variant of this construction that will succeed in constructing $\mathrm{No}_\kappa$, including its map $x \mapsto \omega^x$? Perhaps the answer depends on what is done at limit ordinal steps, but I'm rather confused so I wish someone could clear up the confusion.