In my last company, on the B744, we started 1,2,3 & 4 if possible. System 1 gave us Alternate Brakes whilst system 4 (Aux - and electrical, already selected 'on' before start),) gave us power to Normal Brakes. 2 next as that also supplies alternate brakes. System three is the least important on the ground and #3 was the engine to be shut down first if extended taxying/waiting was expected. #4, (Normal Brakes), would be last as we had not started to move under our own power yet.
Often, at out stations, an engineer unfamiliar with our companies start sequence would clear us, "4,3,2 & 1" but this was normally only in Europe.