[contravariant]

Hmm, turns out this fails to be an ordinal because regular languages aren't well-ordered (except if you reverse the lexicographic order, maybe?). They are what is called an order type, and it looks like it should be possible to identify them with subsets of the rationals, if you wanted to.

Perhaps reversing the lexicographic order makes more sense, in that case longer tuples simply order last so R^* = 1 + R + R^2 + ..., the limit here is much easier since R^*n = 1 + R + ... + R^n is downwards closed as a subset of R^*S(n).

Then again in that scenario [ab]* is simply ω because it is effectively the same as just writing an integer in binary, so it is less interesting in a way.