Define a closure operator on a complete lattice $L$ as a function $f:L \to L$ which is order preserving and idempotent and satisfies $x \leq fx$.Every closure operator arises from an adjunction between $L$ and the lattice of closed elements (those $x$ where $fx = x$). The left adjoint takes $x$ to its closure, $fx$. The right adjoint is the inclusion. Their composite is the closure operator on $L$.
So every closure operator on $L$ arises from an adjunction between $L$ and another lattice. The question is: when does a closure on $L$ arise (as a composite of left and right adjoint) from an adjunction between $L$ and $L$ itself?
It turns out (revising the original question) there are plenty of cases where a closure cannot arise from an adjunction between the lattice and itself. Just switching to interior operators, and asking the same question in the case of $L$ being the powerset, $\mathcal{P}(A)$, of a set $A$ we can see what happens. Use the identification of adjunctions on $\mathcal{P}(A)$ with binary relations on $A$, to show that an interior operation on $\mathcal{P}(A)$ arises from an adjunction iff the join-irreducibles in the lattice of open sets correspond bijectively with a subset of $A$. Of course open here doesn't mean topological open, just with respect to the interior operator. So there is a simple example with $A = \{a,b,c\}$ where the opens are all the subsets except $\{a\}$ and $\{c\}$. This cannot arise from an adjunction because, thinking of a relation as a function $R: A \to \mathcal{P}(A)$, the image of $R$ must have at least the four elements: $\{b\}$, $\{a,b\}$, $\{b,c\}$, and $\{c,a\}$.
I would still be interested to hear of anything more more general than the case of $L = \mathcal{P}(A)$.
The motivation for the question came from the lattice theoretic approach to 'mathematical morphology' in image processing. See for example: L. Najman and H. Talbot (eds), Mathematical Morphology, Wiley, 2010. There is a distinction between 'algebraic closings', which are just general closure operators in the above sense, and 'morphological closings' which are those arising from adjunctions (very often on $\mathcal{P}(\mathbb{Z}^2)$ conceived as a space of images). It's not clear if this term ('morphological closing') is really intended to include examples such as adding pixels to an image if they are not in the image but are surrounded by ones that are.