Denmark has a tradition of private schools with a substantial government subsidy.
This tradition mainly originates in the ideas and initiatives of the clergyman, poet and politician, N.F.S. Grundtvig (1783-1872), and the teacher, Christen Kold (1816-1870). On the basis of their ideas about "a school for life based on the living word", the first "folk high school" for adults was founded in 1844 and the first "free school" (private independent school) for children in 1852. They were in particular meant to serve the rural population.
The ideas of Grundtvig and Kold had such an impact on the political thinking of their time that they were written into the democratic Constitution adopted by Denmark in 1915. It stipulates general compulsory education - not compulsory school attendance.
In Denmark, all children must receive 9 years’ education, but - provided a certain minimum standard is obtained - it is a matter of choice for the parents whether the education is received
1) in the publicly provided municipal primary and lower secondary school,
2) in a private school, or
3) at home.
Full text: http://www.eng.uvm.dk/Education/Primary-and-Lower-Secondary-Education/Private-Schools-in-Denmark