For me this result fits in a context of other results that give complete algebraic invariants for homotopy types. The broad program sometimes goes under the rubric Whitehead's algebraic homotopy program.
If we define a homotopy $n$-type (for $n \geq 1$) as an object of the localization of a suitable category of spaces (e.g., $Top$ or simplicial sets) with respect to maps $f: X \to Y$ that induce isomorphisms on homotopy groups $\pi_k(X, x) \to \pi_k(Y, f(x))$ for each choice of basepoint $x$ and $1 \leq k \leq n$, then it is well-known and classical that homotopy 1-types are classified by their fundamental groupoids, i.e., the localization is equivalent to the category of groupoids. The homotopy hypothesis can be seen as a far-reaching generalization of this basic result; the result is essentially a 1-dimensional truncation of the homotopy hypothesis.
Thus, we could extend this idea of $n$-truncating $\infty$-groupoids past $n = 1$. Homotopy $n$-types are thus classified by $n$-groupoids; it is interesting to see how this subsumes some of the classical results. For example, looking at connected 2-types, these are classified by groupal monoidal groupoids; passing to appropriate skeletal models, this means that connected 2-types $X$ are classified by triples $(\pi_1(X), \pi_2(X), k)$ where $k \in H^3(\pi_1(X), \pi_2(X))$ (an example of a $k$-invariant) is the class of a 3-cocycle
$$\pi_1(X) \times \pi_1(X) \times \pi_1(X) \to \pi_2(X)$$
that in essence specifies an associativity constraint for a monoidal category structure. This description is a modern rendering of work going back to Eilenberg and Mac Lane:
- S. Eilenberg and S. MacLane, Determinationation of the Second Homology and Cohomology Groups of a Space by Means of Homotopy Invariants. Proc. Nat. Acad. Sci., Vol. 32 (1946) 277-280
where the $k$-invariant of a 2-type $X$ can be described in terms of its Postnikov tower.
Along similar lines, Joyal and Tierney (in apparently unpublished work, but circa 1984) described algebraic 3-types in terms of Gray-enriched groupoids, which are appropriately strictified groupoidal tricategories.
Of course, we also know that homotopy types can be described in terms of Kan complexes (which are classical models for $\infty$-groupoids), which is one simplicial manifestation of the homotopy hypothesis. My understanding is that Grothendieck was interested in fundamental algebraic operations on a more globular type of structure that arises from homotopy $n$-types $X$, where the $j$-cells for $j < n$ are maps $D^j \to X$ (thinking here of $D^j$ as a co-globular space) and $n$-cells are maps $D^n \to X$ modulo homotopy rel boundary. Surely this type of structure fits roughly into the sequence of the "classical" weak $n$-categories (category, bicategory, tricategory). If I can indulge in some shameless self-promotion, a main motivation for the notion of weak $n$-category that I presented in 1999 (see Tom Leinster's article) was actually to give a definition of Grothendieck fundamental $n$-groupoids along those sorts of "classical" lines (this is somewhat predating the "$(\infty, 1)$-revolution").
I am not sure of the state of the art here, but if I can be allowed to speculate wildly (and with it being understood that I am not a homotopy theorist): we might begin by observing that objects like Kan complexes (or their truncations) are not, technically speaking, algebraic in the sense of being strictly described in terms of operations subject to equational axioms. They can be algebraized, and something like an algebraic notion of (weak) $n$-groupoid or $\infty$-groupoid results, but actually milking usable algebraic invariants out of such structures, i.e., extracting usable algebraic invariants for $n$-types in a way that extends the results of Eilenberg-Mac Lane, Joyal-Street, etc., seems to me a possibly worthy research project. Possibly such study would proceed by locating usable semi-strictifications of $n$-categorical structures (à la the manner in which Gray-groupoids are semistrict tricategorical groupoids). In other directions: researchers such as Ronnie Brown have worked on $n$-types via more cubical notions (cubical $\omega$-groupoids), especially with a view toward higher van Kampen theorems, and Baues (whose work I don't know) has also worked on Whitehead's program (see his book Algebraic Homotopy, I guess).
While we're in this speculative mode: I don't know what really to say to the suggestion that we generalize the HH to objects of a Grothendieck topos $E$ (whose objects are thought of as "generalized spaces"). The only thing that comes to my mind right away is that we could try defining homotopy groups of objects in $E$ by appealing to a suitable "geometric realization" that passes through a left-exact left adjoint $E \to \textbf{Simp-Set}$. But there could be many choices of such lex left adjoints; they correspond to "$E$-models" in simplicial sets, where model is in the sense of whatever geometric theory the Grothendieck topos $E$ classifies. This could be interesting, or could be a dead end. Hard for me to say.