We explore breather propagation in the damped oscillatory chain with essentially nonlinear (non-linearizable) nearest-neighbour coupling. Combination of the damping and the substantially nonlinear coupling leads to rather unusual two-stage pattern of the breather propagation. The first stage occurs at finite fragment of the chain and is characterized by power-law decay of the breather amplitude. The second stage is characterized by extremely small breather amplitudes that decay hyper-exponentially with the site number. Thus, practically, one can speak about finite penetration depth of the breather. This phenomenon is referred to as breather arrest (BA). As particular example, we explore the chain with Hertzian contacts. Dependencies of the breather penetration depth on the initial excitation and on the damping coefficient on the breather penetration depth obey power laws. The results are rationalized by considering beating responses in a system of two damped linear oscillators with strongly nonlinear (non-linearizable) coupling. Initial excitation of one of these oscillators leads to strictly finite number of beating cycles. Then, the beating cycle in this simplified system is associated with the passage of the discrete breather between the neighbouring sites in the chain. Somewhat surprisingly, this simplified model reliably predicts main quantitative features of the breather arrest in the chain, including the exponents in numerically observed power laws.