Under what conditions can Vbe exceed 0.7
The data sheet will usually give a fair idea about this: -



*Pulse Condition: Pulse Width ≤300μs, Duty Cycle ≤2%.

As you can see, the base-emitter voltage under extreme base currents can be a maximum of 2.6 volts but typically 1 volts is not unheard of.


  In simulation I am observing that Vbe can get a lot larger that 0.7
  volts and Ib is getting large as well. Is this dangerous?


0.7 volts is just a convenient value engineers use. I wouldn't recommend getting too close to the limits and if you need to turn on your 2N2222 a bit harder I would recommend using a MOSFET.