There are many ways to take some real-closed field and generate a proper extension of it with elements that are infinite and infinitesimal relative to the original field. One well-known example is that we can take an ultrapower of the original field. We can then iterate this transfinitely, such that the direct limit of all of these ultrapowers is a proper-class sized extension of the real-closed field. It is a theorem of Ehrlich (and I think earlier of Keisler?) that this field is isomorphic to the surreal numbers in a suitable set theory which can talk about proper classes (typically NBG with global choice), and is maximal in the sense that every real-closed field embeds into it.
One may then ask if we can get the same result iterating other types of construction which are not quite as hardcore as ultrapowers. Here are four, for instance:
If we start with the reals, we can adjoin a new infinitesimal element, called $\epsilon$, and form the field of Newton-Puiseux series in this element. We can then repeat, with the previous field becoming the new field of coefficients, and adjoin another layer of infinitesimals. We can do this for each ordinal, taking the direct limit of previously constructed fields at any limit ordinal, such that the direct limit of the entire thing is a proper class sized real-closed field.
We can do the same thing, but iterating the Levi-Civita construction instead. This is similar, but allows for a slightly more general type of infinite series, and is the Cauchy-completion of the Newton-Puiseux series. This can be transfinitely iterated in the same way.
We can do the same thing with the field of Hahn series $\Bbb R\left[\left[\epsilon^{\Bbb Q}\right]\right]$, which is again, slightly more general than the Levi-Civita field. There are two ways to iterate this: first, we can keep the exponents in $\Bbb Q$ and simply replace the coefficient field with the previously generated Hahn series, as we have previously. Or, perhaps more interestingly, we can even replace the exponents at each step with the previously generated field as well, as well as the coefficients.
Question: do any of these yield the same maximally-sized real-closed field, up to isomorphism, as the surreals and the proper-class sized hyperreals? It would seem that each field is stronger than the previous, but I am wondering if it all shakes out the same way when the construction is iterated to the proper class level. My thinking is that, as ultrapowers aren't necessary to build the maximally sized real-closed field (after all, they aren't necessary to build the surreals), perhaps these weaker constructions also build the same thing if iterated transfinitely.