We report on the experimental observation of the period doubling of multiple dissipative-soliton-resonance (DSR) pulses in an all-normal-dispersion fibre laser, based on a nonlinear amplifying loop mirror. By increasing the pump power, the transition from a single DSR pulse to multiple DSR pulses was observed, in addition to the typical linear pulse broadening, under a fixed pulse peak power. During this process, period doubling appeared because the DSR pulses can exhibit the characteristics of period doubling in a multi-pulse state. The typical DSR performance of a linear pulse duration change, versus the variation of pump power, can be maintained when the period doubling of multiple DSR pulses appears.