Does there exist a category C which such that there is no functor $F:C \rightarrow D$ with $D\not\cong C$ which has a left (or right) adjoint?
The empty category trivially satisfies this (there are no functors at all from a nonempty category to the empty category), but no other such category exists. Let $A$ be any category with a terminal object $1$, and consider the projection $C\times A \to C$. This has a right adjoint $C\to C\times A$ given by $c\mapsto (c,1)$. However, these functors are not equivalences unless $C$ is empty or $A$ is equivalent to the terminal category (and more generally if $C$ is small (or even accessible) it is easy to find an $A$ such that $C\times A$ cannot be equivalent to $C$ by any functor).