I think it really depends on your level of experience, if your starting to bowl a few tournaments...as in local ones, I'd say 4(plastic, light,med,heavy ball)
As you get more advanced, you might keep that type of arsenal, or you may go into a lot more specifics, like balls for specific lines or style, or extremes (eg. bone dry or flooded lanes)
Im sure some of the Pba bowlers would have arsenals of up to 15-20