In many SE posts and the Wikipedia article on the surreal numbers I’ve seen references to a “maximal” hyperreal field that’s isomorphic to the surreals. If they’re isomorphic, then why is it that hyperreal analysis is just as strong as real analysis, but surreal analysis doesn’t even have a functioning exponential function? I’ve also read that there’s another maximal field that has the surreals as a subfield, but I might be misremembering as I’ve also read that the surreals have every other ordered field as a subfield, but I’m not sure of that either.
Could the aforementioned maximal field be enriched by another ultra power construction to produce another field with the former as a subfield?
In Repeated transfer principle for transfinite induction., I read that the ultra power construction of the hyperreals could be repeated an ordinal amount of times to produce another hyperreal field, but does this hold in the “limit” and produce a proper class sized field with the transfer principle, and if so, is it the same as the one mentioned in the Wikipedia article?
I guess I’ve read a bunch of conflicting things and want an answer, but I’d also like resources to read up on to better grasp these concepts as I can’t find any direct sources mentioning this maximal field, ordinally repeated ultrapowers, or the potential field that subsumes the surreals.