maximum principle for norm of gradient of a harmonic function in a riemannian manifold with boundary

Question

Let $(M,g)$ be a compact Riemannian manifold with boundary and $u:M\longrightarrow\mathbb{R}$ a harmonic function. It is known that when $\mathrm{Ric} \geq 0$, $|\nabla u|^2$ is subharmonic due to Bochner formula: $$ \frac 12 \Delta |\nabla u|^2 = |\mathop{\mathrm{Hess}}u|^2 + \mathop{\mathrm{Ric}}(\nabla u, \nabla u)$$ and we conclude that $|\nabla u|^2$ must attain its maximum only in $\partial M$. What can we say about maxima of $|\nabla u|^2$ without assumptions on Ricci curvature? Are there any counterexamples?