MMBT2222 Datasheet doesn't mention maximum base current. How to calculate it?
Base current will contribute to dissipation, which sets a limit (not necessarily the limit). At a maximum Ta of 70°C, according to this [datasheet] the maximum dissipation is 144mW, assuming FR4 and the assumed size and footprints. With 0.7V Vbe, that would limit the base current to about 200mA (assuming zero collector current). However I suspect danger lurks here- the similar MMBT4401 specifies an absolute maximum peak base current of 200mA. So if you substitute an MMBT4401 and keep the base current well below that (say 50mA) and take the total dissipation and ambient temperature maximums into account,  you're probably reasonably safe. 

Normally there is no reason to exceed Ic/10 (and even that is on the high side) with base current so this issue does not arise. Since Ic (max) is 800mA and probably you would not practically use the part over 100-200mA the base current would be in the 5-20mA range, tops. 

If you are attempting something weird like putting the transistor base in series with a relay coil then you may wish to choose a part that at least specifies a limit that you are well within (assuming you care about reliability). And you should also protect the E-B junction against reverse breakdown which can be damaging to the  transistor.