Examples of $\mathcal{O}_X$-modules that are not quasi-coherent sheaves

Examples of $\mathcal{O}_X$-modules that are not quasi-coherent sheaves

Let $X = \operatorname{Spec} k[x]_{(x)}$ which consists of two elements,
the generic point $\zeta$ corresponding to the zero ideal and the closed
point $(x)$. Define an $\mathcal{O}_X$-module $\mathcal{F}$ by setting
$\mathcal{F}(X) = \{0\}$ and $\mathcal{F}(\zeta) = k(x).$ Now
$\mathcal{F}$ is not a quasi-coherent sheaf because if
$\mathcal{F}|_{\operatorname{Spec} k[x]_{(x)}} = \mathcal{F}$ is
isomorphic to $\widetilde{M}$ for some $A$-module $M$, $\mathcal{F}(X) =
0$ implies that $\widetilde{M}(X) = M = 0$. But now $\mathcal{F}(\zeta)$
cannot be isomorphic to $\widetilde{M}(D(0))$ because one is non-zero
while the other is zero. Thus $\mathcal{F} \notin \operatorname{QCoh}(X)$.
Are there any other examples of $\mathcal{O}_X$-modules that are not
quasi-coherent sheaves?