What is the greatest legacy of the Enlightenment?
In fact, the greatest legacy of the enlightenment is the idea of the basic rights of every human being. The above answer correctly notes the influence of the Greeks on democracy, and there is no evidence that the Enlightenment thinkers supported it; but they did believe in the basic rights of every individual.
Typical of this line of thinking is Jean-Jacques Rousseau, in The Social Contract. Rousseau distrusted Democracy, which he considered governance by the masses, but supported the concept of the "general will," which he believed could be determined by a minority which would be comprised of those who were more far-sighted and wiser than the populace at large. The "general will," to Rousseau would meet people's long term needs.
Similarly, the Baron de Montesquieu argued that power came from the people, not from God. He argued for a separation of powers, stating that
it is necessary that by the arrangement of things, power checks power.
John Locke, mentioned above, stated in his Second Treatise on Civil Government that human beings were possessed of certain "natural rights" of life, liberty and property which governments were bound to protect. Should the government be unable or unwilling to protect those rights, then the people had the right to change the government.
Almost all the great Enlightenment thinkers rejected the idea of
Divine Right; and all uniformly espoused the idea that people have certain basic rights which must be preserved. They never suggested that this could be done by democracy. Perhaps their greatest legacy is the preservation of Locke's ideas in the Declaration of Independence and Montesquieu's thoughts in our Constitution which provides for a separation of power.
I would argue that the greatest legacy of the Enlightenment is democracy.
Although democracy of a form did exist in Ancient Greece and in Rome, democracy had disappeared after the fall of Rome. Europe was dominated by monarchies and the idea of a natural hierarchy of people was the basis of every society.
The Enlightenment changed this. Thinkers like John Locke pointed out that monarchies were irrational and that there was no logical reason to believe that some families are naturally meant to rule others. By making this argument, Locke and others like him pointed the way to the sort of democracy that now dominates much of the world. If we accept that democracy is the best form of government, it is the greatest legacy of the Enlightenment.