Tl;DR: four or five
It depends on the servos you're using and the battery chemistry (alkaline batteries usually have a higher cell voltage than rechargeable cells like NiCd, NiMH). I believe that the servos we ship with the module are rated to 6V, so 4 AA batteries at 1.2-1.5V each might be more appropriate. 6- and 8-battery packs might provide too high of a voltage, which could damage the servo electronics. That said, these devices have very wide tolerances, so the only way to know for sure is to try (and risk burning out one servo).
If one battery pack doesn't cut it, you can wire two identical packs in parallel to give the servos access to more current and a lower effective series resistance, both of which can help get more torque out of a motor.