I think you all should to think about this question differently... what is the most influential advancement, that was not destined to happen anyways? Lanuage was inevitable, as was math, the wheel, and agriculture. But some things, like car engines, microchips, medical technologies, the space shuttle, nuclear weapons, etc. were not such sure things. That's where the question gets interesting. Because if you just go by what was the most important, well then it is fire.. because without knowing how to make fire we'd all be dead.
I personally would go with nuclear weapons, because of the way they completely reshaped the interaction of nations and how one day we will see big things come from them, whether they be good or bad (so far it's been good).