  We propose and analyze an $H^2$-conforming Virtual Element Method (VEM) for
the simplest linear elliptic PDEs in nondivergence form with Cordes
coefficients. The VEM hinges on a hierarchical construction valid for any
dimension $d \ge 2$. The analysis relies on the continuous Miranda-Talenti
estimate for convex domains $\Omega$ and is rather elementary. We prove
stability and error estimates in $H^2(\Omega)$, including the effect of
quadrature, under minimal regularity of the data. Numerical experiments
illustrate the interplay of coefficient regularity and convergence rates in
$H^2(\Omega)$.
