What is the smallest subfield $F\subset N_0$ such that $$(F,+,\times,\leq)\ncong(N_0,+,\times,\leq)$$ but $$(F,+,\leq)\cong(N_0,+,\leq)?$$ Since these are all going to be proper classes cardinality is not sufficient to delineate what we mean by 'smallest', so we use the following notion. For two proper class sized ordered fields $F,F'$ we will say that $F$ is *smaller* than $F'$ iff there exists an injective order preserving homomorphism of the natural value group of $F$ into the natural value group of $F'$, but the reverse does not hold.

I asked this question to a friend recently and he pointed out that the class of all Conway normal forms with rational coefficients satisfies the above isomorphism identities, and since the natural value group of an ordered field is determined by the multiplicative structure of the field I believe this means that we will have such a one sided injection between their value classes. He also pointed out that the group $\mathbb{J}$ of all purely infinite surreal numbers is the classical example of a subgroup of $N_0$ which is isomorphic to it as an ordered group but not as a field, but this structure does not form a field as it does not contain $1$.

Are there any subfields 'smaller' than this with the desired property? Can we determine what the 'smallest' one is up to an appropriate type of morphism?