Motivation for relative schemes: why should one work with schemes over a ringed topos? Recently I've been trying to learn more about relative schemes. These were developed in M. Hakim's thesis Topos annelés et schémas relatifs under Grothendieck's guidance and appear in many of later works of the Grothendieck school, such as Berthelot's Cohomologie Cristalline des Schemas de Caracteristique $p>0$ or Illusie's Complexe Cotangent et Déformations I et II.

Question I. What are some instances in which working in the full generality of a ringed topos gives one more powerful tools than just working with $S$-schemes?

One example I'm aware of is in Illusie's Complexe Cotangent books. As remarked by Jonathan Wise in this MO question, working with ringed topoi in this setting enables one to study more interesting deformations.
I heard that a modern example might be the Falting topos, which appears in Abbes–Gros–Tsuji's book The p-adic Simpson Correspondence. (I don't really understand this example, however.)
 A: The comments to your question discuss the variation of relative schemes over a topos, vs relative schemes over a site. But it seems your question 
stood at the more basic level of the relevance of relative schemes over something else than a scheme.
In Grothendieck's philosophy, a relative scheme $f\colon X\to S$ functions as a family of schemes $X_s$ (for $s\in S$) parameterized by the base $S$, which is a scheme. Both $X$ and $S$ are schemes, and $f$ is a morphism of schemes (possibly with additional properties such as being flat, proper, finite, étale, smooth, etc.)
There are instances where one wants to consider families of schemes which are parameterized by something else, such as a (complex, Berkovich, Huber…) analytic space.  For example for formulating GAGA-type theorems: what do complex analytic families of complex varieties in a given projective space look like, when the parameter set is an open disk, say.
In most important cases, those spaces are essentially characterized by rings (eg, local rings, or rings of functions over a compact, affinoid subspace) and sometimes the above study can be reduced to the study of relative schemes over these rings. Such a technique is systematically used in nonarchimedean geometry, for instance.
Nevertheless, it might be interesting to have under disposal a full-fledged theory of relative schemes over bases.
Being over a ringed topos, Monique Hakim's theory can encompass all of the above situations.
In any case, such a theory won't prove the basic (but difficult) results from commutative algebra that are probably needed. In nonarchimedean geometry, some work is needed, for example, to compare the scheme $\mathop{\rm Spec}(A)$ and the affinoid space $\mathscr M(A)$, when $A$ is an affinoid algebra, and similarly for a relative family $X\to \mathop{\rm Spec}(A)$ and its analytification $X^{\mathrm {an}}\to \mathscr M(A)$.
