If the ground prong on my extension cord breaks off, can I still plug the extension cord into an outlet without danger?
The ground prong on an electric cord is there to protect you in case something goes wrong with the tool or appliance while you are handling or touching it. You will not die immediately without the ground prong, but you will no longer be protected from shock, burns, or electrocution by a faulty device.
There are several things that can happen to energize the casing of an electric device. The most likely is that the insulation on the hot wire could fail, causing the hot to touch the casing. This will cause the circuit breaker to pop open and protect you -- but only if the device is properly fault grounded. The ground prong is an essential part of this fault grounding.
I once heard my neighbor's kid complaining about the rules imposed by his mother. His exact words were "She's so dumb. I played in the street hundred of times and I never got hit by no bus." Operating without the ground prong is playing in the street and hoping there's no bus.
If the tool is undamaged and used in a safe manner, it will work. You will not be protected if either of those assumptions is false.