My question is as in the title:

Does anyone have an example (supposing one exists) of an $\infty$-topos which is known not to be equivalent to sheaves on a Grothendieck site?

An **$\infty$-topos** is as in Higher Topos Theory (HTT) 6.1.0.4: an $\infty$-category which is an accessible left-exact localization of presheaves on a small $\infty$-category.

A **Grothendieck site** is a small $\infty$-category $\mathcal{C}$ equipped with the $\infty$-categorical variant of the classical notion of a Grothendieck topology $\mathcal{T}$, as in HTT 6.2.2: a collection of **sieves** (subobjects $U\to j(C)$ of representable presheaves on $\mathcal{C}$) satisfying some axioms. **Sheaves** on $(\mathcal{C},\mathcal{T})$ are presheaves of $\infty$-groupoids on $\mathcal{C}$ which are local for the sieves in $\mathcal{T}$. Such form a full subcategory $\mathrm{Shv}(\mathcal{C},\mathcal{T})$ of the $\infty$-category of presheaves.

Note: the question Examples of $(\infty,1)$-topoi that are not given as sheaves on a Grothendieck topology appears superficially to be equivalent to this one. In practice it is not exactly the same. As answers to that question show, many interesting $\infty$-topoi exist which can be described without reference to any Grothendieck site. But it is still conceivable that a suitable site exists.

Also note: any $\infty$-topos $\mathcal{X}$ can be obtained as an accessible left-exact localization of some $\mathrm{Shv}(\mathcal{C},\mathcal{T})$ with respect to a suitable class of $\infty$-connected morphisms (HTT 6.2.2, 6.5.3.14), e.g., the class of hypercovers. However, this does not immediately preclude $\mathcal{X}$ being equivalent to $\mathrm{Shv}(\mathcal{C}',\mathcal{T}')$ for some other Grothendieck site $(\mathcal{C}',\mathcal{T}')$.

**Added remark.** I asked this question because I had, for a long time, tacitly assumed that such examples were plentiful, until I thought about it and realized I had no basis for thinking that. As no answers have yet been given, and I'm not aware of any tools which would likely lead to a resolution one way or the other, it looks to me that this should be regarded as an open question.

thinkwhat happens is that if you try to generalize the proof, you learn that $\infty$-topoi are left-exact localizations of presheaves on themselves. The issue then is that left-exact localizations need not come from Grothendieck topologies in the $\infty$-case, i.e., need not be "topological localizations" in the sense of HTT 6.2.1.4, which are characterized as being determined by the class ofmonomorphismswhich are inverted. $\endgroup$hypercompletetopos is the hypercompletion of a sheaf topos. $\endgroup$4more comments