  For strongly monotone dynamical systems on a Banach space, we show that the
largest Lyapunov exponent $\lambda_{\max}>0$ holds on a shy set in the
measure-theoretic sense. This exhibits that strongly monotone dynamical systems
admit no observable chaos, the notion of which was formulated by L.S. Young. We
further show that such phenomenon of no observable chaos is robust under the
$C^1$-perturbation of the systems.
