The effect of gas compression at the developed stages of flame acceleration in smooth-wall and obstructed channels is studied. We demonstrate analytically that gas compression moderates the acceleration rate, and we perform numerical simulations within the problem of flame transition to detonation. It is shown that flame acceleration undergoes three distinctive stages: (1) initial exponential acceleration in the incompressible regime, (2) moderation of the acceleration process due to gas compression, so that the exponential acceleration state goes over to a much slower one, (3) eventual saturation to a steady (or statistically steady) high-speed deflagration velocity, which may be correlated with the Chapman-Jouguet deflagration speed. The possibility of deflagration-to-detonation transition is demonstrated.