The Hundred Years War was probably the very best chance of this happening, especially if Henry V had survived longer as everyone has pointed out. I started thinking about what could have happened had Edward IV had not married Elizabeth Woodville and married the French princess Warwick was trying to arrange him with, eschewing Edward's preferred Burgundian alliance instead of a French one. Then I realized that all that would happen would be even more intertwining between the prevailing English and French Royal houses and the possibility of yet another dynastic squabble down the line, except now with better technology and deeper international agreements possibly spiraling out of control into an even larger war.