For a brief period from 1790-1791, France was a constitutional monarchy, with Louis XVI having sworn an oath to uphold the France's new (if yet unrealized) Constitution. Roughly a year later, the King attempted the infamous and ill-fated Flight to Varennes, which pretty much sunk what support the Bourbons still enjoyed.
So, what if France had remained a constitutional republic? Obviously no Napoleonic Wars, but what else might have happened?