Let's treat $\mathbf{No}$ as a group under addition, and forget its field structure for a little bit.

I will define a "maximally Archimedean subgroup" of $\mathbf{No}$ as a subgroup which is

- Archimedean, and
- not strictly contained by any larger Archimedean subgroup of $\mathbf{No}$.

Each of these groups is isomorphic to $\mathbb{R}$. Examples: $\mathbb{R}$, $\omega\mathbb{R}$, $\frac{1}{\omega}\mathbb{R}$, $\sqrt{\omega_1}\mathbb{R}$ etc.

Now consider the direct sum of all of these subgroups. **Is this group isomorphic to $\mathbf{No}$?**

Also, can anything useful be said about the direct product of the subgroup instead, other than that it contains things like the sum of all infinite cardinals, stuff like $\sum_{r\in\mathbb{R}} \omega^r$, $\sum_{r\in\mathbf{Ord}} \omega^r$, $\aleph_0+\aleph_1-\aleph_2+\aleph_3-...$, etc, and hence is just baffling all around?

You can formalize $\mathbf{No}$ however you like when dealing with the surreals to handle the foundational issues.