If topology were invented for algebraic geometry or logic, in ignorance of Euclidean space, we might reasonably regard connected compact Hausdorff spaces as pathological, or even doubt their existence. After all, Hausdorffness is a "separation" condition whereas connectedndess is a "nonseparation" condition -- it would seem strange that they should be compatible if you'd only seen spectral spaces or Stone spaces but not Euclidean spaces.
Luckily though, we know about Euclidean space and spaces built from it, and their central importance to mathematics. But they are very special, being as they are balanced between separatedness and nonseparatedness. For instance, every connected compact abelian group is a torus. Every number field has finitely many infinite places -- which are mysterious in a way that finite places are not. And so forth.
When it comes to $\infty$-topos theory as a generalization of topology, I wonder what the infinite places are. All the generalized concepts are there:
A proper geometric morphism (HTT 18.104.22.168) $\mathcal Y \to \mathcal X$ is one which satisfies a Beck-Chevalley condition, stably under pullback.
So a compact $\infty$-topos $\mathcal X$ is one such that the global sections geometric morphism $t_\ast: \mathcal X \to Spaces$ is proper.
And a Hausdorff $\infty$-topos $\mathcal X$ is one such that the diagonal $\mathcal X \to \mathcal X \otimes \mathcal X$ is proper (here $\mathcal X \otimes \mathcal X$ is the Lurie tensor product, which is the product in the category of toposes by HTT 22.214.171.124).
Generalizing from ordinary topos theory, an $\infty$-connected geometric morphism is one such that the left adjoint $t^\ast: Spaces \to \mathcal X$ to global sections is fully faithful.
Generalizing again from ordinary topos theory, a locally $\infty$-connected topos is one such that $t^\ast$ has a further left adjoint $t_!: \mathcal X \to Spaces$.
What's an example of a compact Hausdorff, $\infty$-connected, locally $\infty$-connected $\infty$-topos other than $Spaces$ itself?
There are various ways to relax these conditions:
If we don't require $\infty$-connectedness, then any étale topos (i.e. a slice $Spaces/X$) satisfies the remaining conditions; I'd be interested in examples other than these.
If we don't require Hausdorffness, then we have a weakening of the notion of a cohesive topos (i.e. a topos which is local (a strengthening of compactness) and strongly connected (a strenghtening of connected + locally connected)). My sense is that there are lots of cohesive toposes, so I think that Hausdorffness may be what sets this question apart from questions about cohesion.
We could relax compactness to local compactness or exponentiability or even do away with it altogether. I'd be interested in such examples.