What is a simple example of a topological space $X$ and a complete category $\mathcal{C}$ such that the inclusion $\mathbf{Sh}(X,\mathcal{C}) \hookrightarrow \mathbf{PSh}(X,\mathcal{C})$ has no left adjoint, i.e. there is a $\mathcal{C}$-valued presheaf on $X$ which does not have an associated sheaf?
A sufficient condition for the existence of the associated sheaf is explained in Section 17.4 in Categories and sheaves by Kashiwara-Schapira: here the category $\mathcal{C}$ is assumed be be complete, cocomplete, filtered colimits are exact, and the IPC property holds. So for a counterexample one of these properties must fail.