Let $R$ be a discrete valuation ring and $V$ a free $\mathcal O_X$-module of infinite rank on $X=\mathrm{Spec}(R)$. Then, neither $\mathcal Hom(V,\mathcal O_X)$ nor $\mathcal Hom(\mathcal Hom(V,\mathcal O_X),\mathcal O_X)$ are is quasi-coherent. In particular, $V$ is not reflexive.
Let $R$ be a discrete valuation ring and $V$ a free $\mathcal O_X$-module of infinite rank on $X=\mathrm{Spec}(R)$. Then, neither $\mathcal Hom(V,\mathcal O_X)$ nor $\mathcal Hom(\mathcal Hom(V,\mathcal O_X),\mathcal O_X)$ are quasi-coherent. In particular, $V$ is not reflexive.