Probably the "Battle of Long Island", (or the "Battle of Brooklyn", or the Battle for New York) in 1776.

This battle was a defeat for Washington and the Revolutionary army, but it easily could have been an absolute disaster. Washington was defending New York, with about 10,000 men, while the British were attacking with 30,000. Both sides bungled badly. Washington made an almost fatal error by deploying his troops on Long Island, where British Seapower could have cut him off from the American mainland and led to his inevitable defeat. The British bungled badly by not moving quickly and aggressively enough to cut him off.

Washington managed to evacuate his whole army by night across the East River to Manhattan on boats manned by the Marblehead Regiment. Then, he managed to out march the British (just) North up Manhattan to escape into mainland New York State. If the British fleet had pushed up the Hudson earlier, they could have cut him off.

If Washington and the entire Continental Army had been cut off on Long Island or Manhattan, and forced to surrender, that might well have been the end of the Revolution. It certainly would have been difficult for the Revolutionaries to fight on.

Washington saw the trap in the nick of time, and managed to evade it (helped by some luck and some British bungling). He was a fast learner, and was careful to never again put himself in a position where he could lose the whole war in a day, or be cut off from the vast American hinterland.

This battle was the best military chance for the British to win the Revolutionary War, and they came close. Washington's escape from the trap was perhaps essential for eventual Revolutionary victory.