For any space $X$, there's an $\infty$-topos of spaces fibered over $X$. The underlying
ordinary topos is the category of representations of the fundamental groupoid of $X$.
So if $X$ is simply connected, this is just the category of sets. But the $\infty$-topoi are different for different values of $X$ (two spaces $X$ and $Y$ yield equivalent $\infty$-topoi if and only if $X$ and $Y$ are homotopy equivalent).
I will write what I think is a proof that in fact every infinity topos is equivalent to sheaves over itself. Please let me know if I am making any errors. I am basically adapting a proof from SGA4 of the classical statement.
Let $U$ a Grothendieck universe and suppose that $E$ is a left-exact localization of presheaves of $U$-small infinity groupoids on some $U$-small site. Then $E$ possesses a $U$-small set of generators, $X_\alpha$, $\alpha \in A$. By HTT 6.3.5.17, the Yoneda embedding of $E$ into sheaves of $V$-small infinity groupoids on $E$, with $U \in V$ a larger Grothendieck universe, preserves $U$-small colimits.
Lemma: If $i:F \hookrightarrow G$ is a mono with $F$ and $G$ infinity sheaves on $E$, with $G$ representable, then $F$ is representable.
Pf: Consider the family $\left(f:X_\alpha \to F, f \in F\left(X_\alpha\right)_0\right), \alpha \in A$, which is jointly epimorphic. Hence, the corresponding Cech-nerve is effective. Each iterative fiber product in this nerve, say $$X_{\alpha_1} \times_F ...\times_{F} X_{\alpha_n}\simeq X_{\alpha_1} \times_G ...\times_{G} X_{\alpha_n},$$ since $i$ is mono. But the right-hand side is representable. Hence, this is actually an effective groupoid object in $E$, so it has a colimit $C$ in $E$. Since the inclusion of $E$ into $V$-sheaves (the Yoneda embedding) preserves colimits, we conclude that $C$ represents $F$.
Now consider $H$ to be an arbitrary sheaf of $U$-small infinity groupoids on $E$. Consider the family $\left(g:X_\beta \to H, f \in H\left(X_\beta\right)_0\right), \beta \in B$. Each iterative fiber product in the associated nerve, say $X_{\alpha_1} \times_H ...\times_{H} X_{\alpha_n}$ is a subobject of the representable sheaf $X_{\alpha_1} \times...\times X_{\alpha_n}$, but also an infinity sheaf, hence it is representable by an object of $E$ by the lemma. So we again have an effective groupoid object which actually lies entirely in $E$, and as in the lemma, we conclude $H$ is representable.
Best Answer
No. Some, but not all topoi are exponentiable.