Marc's examples are good ones, but let me add two more (which are closely related to each other):

1) Let $\mathcal{C}$ be an accessible $\infty$-category which admits small filtered colimits, and let $\mathcal{X}$ be the $\infty$-category
of functors from $\mathcal{C}$ to $\mathcal{S}$ which preserve small filtered colimits. Then $\mathcal{X}$ is an $\infty$-topos, but I do not know how to realize $\mathcal{X}$ as an $\infty$-category of sheaves on a site.

2) Let $X$ be a locally compact topological space, and let $\mathcal{X}$ be the $\infty$-topos of sheaves on $X$. Then $\mathcal{X}$ is exponentiable in the setting of $\infty$-topoi: that is, for every $\infty$-topos $\mathcal{Y}$, there exists another $\infty$-topos $\mathcal{Y}^{\mathcal{X}}$ such that geometric morphisms from $\mathcal{Z}$ into $\mathcal{Y}^{\mathcal{X}}$ are the same as geometric morphisms from $\mathcal{Z} \times \mathcal{X}$ into $\mathcal{Y}$
(where the product is formed in the $\infty$-category of $\infty$-topoi). I do not know if $\mathcal{Y}^{\mathcal{X}}$ can be realized as an $\infty$-topos of sheaves on a site, even if it is assumed that such a description is known for $\mathcal{Y}$.