The current gain of the 2n2222A is 40 @ 500mA Ic, refer to the datasheet here http://www.uib.es/depart/dfs/GTE/staff/jfont/InstrETT/2n2222.pdf
0.3A/40 = .0075A of base current needed. Double this number, because a little extra won't hurt.
so then you take the 555 output voltage, which you have...