What is the "right" surreal generalization of the fact that a real number $r$ is rational if and only if its sign-expansion is eventually periodic?
I can think of more than one natural way to generalize the notion of a rational number, but "ratio of omnific integers" is not one of them, since every real number is a ratio of omnific integers. Maybe ratios of ordinals are the thing to look at (as a generalization of the non-negative rationals). Or maybe we should look at the Field-closure of the ordinals. Or perhaps we should consider the Class containing every surreal number whose normal form involves only rational numbers at all levels.
I can also think of more than one way to generalize the notion of eventual periodicity to sign-sequences indexed by a general ordinal alpha. One of them is a variant of "Kaufman decimals" (see https://mchouza.wordpress.com/2013/08/25/kaufman-decimals/ and http://www.jefftk.com/p/decimal-inconsistency) in which the digit-set {0,...,9} is replaced by {$+$,$-$} and every over-bar is assigned an ordinal.
A seemingly different but possibly equivalent notion generalizing eventual periodicity involves a kind of symbolic dynamics I haven't seen before, where the Monoid of ordinals acts on ordinal-indexed sequences: if $s$ is a sequence indexed by some initial segment of the ordinals, and $\iota$ is some ordinal, define $T^{\iota}(s)$ to be the sequence obtained by omitting the first $\iota$ terms of $s$ (with $T^{\iota}(s)$ defined to be the empty sequence if $\iota$ is greater than or equal to the length of $s$). Then eventual periodicity (in the case where $s$ is indexed by the natural numbers) is seen to be a special case of the condition that the orbit of $s$ under the action of the Monoid of all ordinals is finite. (See Joel Hamkins' recent post, showing that constant sequences satisfy this finiteness condition: http://jdh.hamkins.org/every-ordinal-has-only-finitely-many-order-types-for-its-final-segments/.)
If my original question seems too vague (what does "right generalization" mean?), here are two very concrete ones that are relevant: does the surreal number with sign-expansion $+-^{\omega}++-^{\omega}+++-^{\omega}++++-^{\omega}\cdots$ (indexed by $\omega^2$) lie in the field generated by $\omega$? and, is it expressible as a ratio of ordinals? (Note that this sign-sequence does not satisfy the aforementioned finiteness property, although perhaps it satisfies some weaker regularity condition.) Here $-^{\omega}$ denotes a string of $\omega$-many $-$'s and $\cdots$ denotes that the pattern continues $\omega$ times.
I'd be interested in any implications that might hold between these various properties.