2N2222 Base Current Maximum Tolerance
The Fairchild PN2222 data sheet says the base-emitter saturation voltage is 2V with a collector current of 500mA and a base current of 50mA. This is for a small duty cycle of 2% with the pulse width less than 300us.

I guess this means the base could take 50mA but how long could it take it for is a bit of a guess really. 

If we look at the wattages, 50mA x 2V = 100mW and with collector-emitter saturating at 1V for Ic = 500mA we get 500mW. That's 600mW in total and pretty close to the absolute max rating of 625mW.

You could argue that if the collector current were limited to somewhat less than 500mA the overall power disippation would be less and maybe the 50mA is sustainable indefinitely.

Multicomp's 2N2222 data sheet quotes min H\$_{FE}\$ of 30 at a collector current of 500mA and this implies a base current of 16.67mA. The same data sheet also quotes 50mA base current when specifying base-emitter saturation voltage. It also has the same caviats on duty cycle and pulse width as Fairchild.

It looks like ON semi's data sheet has a nice graph on page 4 (figure 4) that gives a bit more detail.

My personal conclusion - don't run the base current continuously at more than 20mA - this is based on gut feeling and the data I've seen.