The short answer is no. Even 2-toposes are poorly understood -- we don't know what the right definition is. For higher dimensions, including $\infty$, we definitely don't have the answers.
Just as the primordial example of a (1-)topos is $\mathbf{Set}$ (the 1-category of sets and functions), the primordial example of a 2-topos should be $\mathbf{Cat}$ (the 2-category of categories, functors and natural transformations).
Mark Weber has done some work on 2-toposes, building on earlier ideas of Ross Street. But I think Mark is quite open about the tentative nature of this so far.
There was a good discussion of the current state of 2-toposes (and more generally n-toposes) at the $n$-Category Café last year:
Suppose $k$ is a field, not necessarily algebraically closed. $\text{Spec } k$ fails to behave like a point in many respects. Most basically, its "finite covers" (Specs of finite etale $k$-algebras) can be interesting, and are controlled by its absolute Galois group / etale fundamental group. For example, $\text{Spec } \mathbb{F}_q$, the Spec of a finite field, has the same finite covering theory as $S^1$, which reflects (and is equivalent to) the fact that its absolute Galois group is the profinite integers $\widehat{\mathbb{Z}}$. (So this suggests that one can think of $\text{Spec } \mathbb{F}_q$ itself as behaving like a "profinite circle.")
More generally, suppose you want to classify objects of some kind over $k$ (say, vector spaces, algebras, commutative algebras, Lie algebras, schemes, etc.). A standard way to do this is to instead classify the base changes of those objects to the separable closure $k_s$, then apply Galois descent. The topological picture is that $\text{Spec } k$ behaves like $BG$ where $G$ is the absolute Galois group, $\text{Spec } k_s$ behaves like a point, or if you prefer like $EG$, and the map
$$\text{Spec } k_s \to \text{Spec } k$$
behaves like the map $EG \to BG$. In the topological setting, families of objects over $BG$ are (when descent holds) the same thing as objects equipped with an action of $G$. The analogous fact in algebraic geometry is that objects over $\text{Spec } k$ are (when Galois descent holds) the same thing as objects over $\text{Spec } k_s$ equipped with homotopy fixed point data, which is a generalization of being equipped with a $G$-action which reflects the fact that $k_s$ itself has a $G$-action.
(I need to be a bit careful here about what I mean by "$G$-action" to take into account the fact that $G$ is a profinite group. For simplicity you can pretend that I am instead talking about a finite extension $k \to L$, although I'll continue to write as if I'm talking about the separable closure. Alternatively, pretend I'm talking about $k = \mathbb{R}, k_s = \mathbb{C}$.)
The classification of finite covers is the simplest place to see this: the category of finite covers of $\text{Spec } k_s$ is the category of finite sets, with the trivial $G$-action, so homotopy fixed point data is the data of an action of $G$, and we get that finite covers of $\text{Spec } k$ are classified by finite sets with $G$-action.
But Galois descent holds in much greater generality, and describes a very general sense in which objects over $k$ behave like objects over $k_s$ with a Galois action in a twisted sense.
Best Answer
Here's an example which, I'm afraid, is not very interesting (and may not match your notion of “suitable sense”): let $X$ be the topological space $\mathopen]0,1\mathclose[$ (the open interval) with the (not at all separated) topology given by the $U_x := \mathopen]0,x\mathclose[$ for $x \in [0,1]$ (with $U_0 = \varnothing$, obviously). Since $U_x \cap U_y = U_{\min(x,y)}$ and $$ \bigcup_{i\in I} U_{x_i} = U_{\sup\{x_i : i\in I\}} $$ this is indeed a topology. Of course, $\mathcal{O}(X) := \{U_x : x\in[0,1]\}$ can be identified with $[0,1]$ as an ordered set (indeed, frame). The topos of sheaves on $(X, \mathcal{O}(X))$ then has this as subobject classifier.