In order to analyze the maximum Raman gain of forward-pumped and backward-pumped fibre Raman amplifier(FRA), by theoretical derivation method for solving the differential coupling equations, the formulas for the maximum Raman gain of forward-pumped and backward-pumped FRA were defined and obtained based on different signal amplification process with different pumping structure. Using these new formulas, the effects of input signal power, input pump power, Raman gain coefficient per area, fibre attenuation and the pump to signal frequency ratio on the maximum Raman gain of forward-pumped and backward-pumped FRA were investigated in detail in the condition that the parameters were all of their typical values. Some results are consistent with results of involved references. Besides, maximum Raman gains of two pumping ways under the same parameter were compared. Finally, maximum Raman gain and on-off gain which was frequently-used were compared. These results are creative and helpful to the experiment study of FRA.