I am also not sure what initial poster means exactly by free public education. One of early examples was Polish-Lithuanian Commonwealth. Country was reasonably tolerant to different religions, so when Protestant ideas begun to spread both Catholics and Protestants were trying to win "Hearts and minds" by building free schools. This effort was expanded to all people including peasants. In 1773 Commonwealth was first country in Europe that created something like Ministry of education. This was also connected with religious upheavals. A lot of schools including universities were owned by Jesuits, when order was disbanded all its property was transferred for maintaining their schools by the state. This might have had huge impact, but Commonwealth was destroyed shortly after and this reform didn't have time to bear fruit.