Many of these toposes admit descriptions as internal classifying toposes, hence indeed enjoy useful universal properties. Here is a selection of such descriptions:

*Constructing the big Zariski topos from the little Zariski topos.* The big Zariski topos of a scheme $S$ is the externalization of the result of constructing, internally to the little Zariski topos of $S$, the classifying topos of local $\mathcal{O}_S$-algebras which are local over $\mathcal{O}_S$.
(One might hope that it would simply be the internal big Zariski topos of $\mathcal{O}_S$, that is the classifying topos of local $\mathcal{O}_S$-algebras where the structure morphism needn't be local. This alternative description is true if $S$ is of dimension $0$.)

*Constructing the little Zariski topos from the big Zariski topos.* Additionally to $\mathbf{A}^1$, which is the functor $T \mapsto \Gamma(T,\mathcal{O}_T)$, the big Zariski topos contains an additional local ring object: The functor $\flat \mathbf{A}^1$, which maps an $S$-scheme $(f : T \to S)$ to $\Gamma(T,f^{-1}\mathcal{O}_S)$. There is a local ring homomorphism $\flat \mathbf{A}^1 \to \mathbf{A}^1$, and the little Zariski topos can be characterized as the largest subtopos of the big Zariski topos where this morphism is an isomorphism. (This fits with the comments as follows. If $S = \operatorname{Spec}(A)$ is affine, we have the string of ring homomorphisms $\underline{A} \to \flat \mathbf{A}^1 \to \mathbf{A}^1$ starting in the constant sheaf $\underline{A}$. The map $\underline{A} \to \flat \mathbf{A}^1$ is always a localization, and the composition is iff $\flat \mathbf{A}^1 \to \mathbf{A}^1$ is an isomorphism.)

*Constructing the big étale topos from the big Zariski topos.* The big étale topos of a scheme $S$ is the largest subtopos of the big Zariski topos of $S$ where $\mathbf{A}^1$ is separably closed. This fact is essentially a restatement of Gavin Wraith's theorem on what the big étale topos classifies.

*Constructing the big infinitesimal topos from the little Zariski topos.* A back-of-the-envelope computation indicates that the recent result of Matthias Hutzler on what the infinitesimal topos of an affine scheme classifies can be relativized to the non-affine case as follows. The big infinitesimal topos of a scheme $S$ is the externalization of constructing, internally to the little Zariski topos of $S$, the classifying topos of local and local-over-$\mathcal{O}_S$ $\mathcal{O}_S$-algebras equipped with a nilpotent ideal.

Some details can be found in Sections 12 and 21 of these notes.

A word of warning: When I say "the largest subtopos where *foo*", I refer to the largest element in the poset of subtoposes which validate *foo* from their internal language. (By general abstract nonsense (more or less the existence of classifying toposes), such a largest element always exists in case *foo* is a set-indexed conjunction of geometric implications.) In particular, I'm *not* referring to "the subtopos of those objects $Y$ of $E$ such that $\mathbf{A}^1$ restricted to $Y$ enjoys *foo*" (as in the comments). Indeed, this category is in general not a subtopos (typically it doesn't contain the terminal object). Maybe I interpreted that phrase too literally.

notwhat we might expect: Their ring-theoretic parts are required to be isomorphisms instead of local homomorphisms (as would be the case in the category of locally ringed spaces). The Spec functor is indeed an adjoint, if we let it map to the category of locally ringed toposes (objects are pairs $(\mathcal{E},\mathcal{O}_\mathcal{E})$, morphisms are pairs $(f:\mathcal{E}\to\mathcal{F}, f^\sharp:f^{-1}\mathcal{O}_\mathcal{F}\to\mathcal{O}_\mathcal{E})$), see for instance Sect. 12 of these notes. $\endgroup$