In set theory, one can use a non-principal ultra filter, $U_0$, to construct the hyperreal numbers. $\mathbb{R}^*$ is constructed as $\mathbb{R}^{\mathbb{N}}/U_0$. The transfer principal means that first-order properties are transferred over to the new object. However, this construction can be repeated to make $\mathbb{R}^{*2} \equiv (\mathbb{R}^*)^{\mathbb{N^*}}/U_1 $, where $U_1$ is a non-principal ultra filter on the hyperreals. The same transfer principal applies and this set is also a field with the hyperreals as a subfield.
This can be repeated with induction proving that the transfer principal applies to each one: $\mathbb{R}^{*i}\equiv (\mathbb{R}^{*(i-1)})^{\mathbb{N}^{*(i-1)}}/U_{i-1}$, where $U_{i-1}$ is a non-principal ultra filter on $ \mathbb{R}^{*(i-1)}$. And the base case is trivial (all first-order—-and other—- properties of $\mathbb{R}$ apply to $\mathbb{R}$).
However, I don’t know how to go from that to the transfinite case where $i$ is a limit ordinal like $\omega$. My original thought was to use something like a direct limit to construct $\mathbb{R}^{*\omega}$, but I don’t know if the transfer principal still follows. If it does apply for every limit ordinal however, then we could make a field that is a proper class from NBG (like the surreals), but it still has the transfer principal.
In case anyone wants to know why I’ve explored this territory, it’s because I got disappointed when learning that the surreal numbers couldn’t actually be used for any serious analysis and got more interested in the hyperreals instead. As I learned the ultra filter construction, I wondered if the process could be repeated and a stack exchange answer I'd seen had confirmed it. From there, I thought of repeating it indefinitely and seeing if the hyperreals could achieve the same thing that made the surreals interesting: it’s cardinality.
In fact, if this theoretical proper class-sized field could be made, then the construction could be continued until it becomes a conglomerate from MK set theory and maybe even further, but I don’t know if any set theory that supports such an idea(but I could probably make something work like the category of “$n$-set”, just like how higher-order categories work).
Yes, this works: more generally, if $(\mathcal{A}_i)_{i<\lambda}$ is a sequence of structures with $\mathcal{A}_i\preccurlyeq\mathcal{A}_j$ whenever $i<j$ and $\lambda$ is a limit ordinal, then we get $\mathcal{A}_i\preccurlyeq\mathcal{B}:=\bigcup_{i<\lambda}\mathcal{A}_i$ as well. The relevant phrase to look for is "(union of an) elementary chain."