Is there any [industrialized, first-world, use whatever term you want to use] nation where education is not compulsory OR where free education is not provided by the government (or both)?
The Cayman Islands do not provide free government schooling to non-citizens legally resident in the country. That is a sizable chunk of the youth population considering about 50% of the workforce are not citizens.
Schooling is, however, mandatory. Expats must pay for private schools for their kids.
In Ireland (south) education is not, for the most part,* provided by the government. It is provided by other groups such as churches, Irish language organisations and more recently a body called Educate Together which was established for the specific purpose of setting up multi-denominational/integrated schools.
The government subsidises them, so they are “free” in terms of enrolment/tuition (although the subsidy is so small that parents end up being pressured into donations, and things like books and mandatory uniforms have to be paid for by parents too). The government also sets a national curriculum, but schools have a certain amount of leeway in how it is implemented.
Education is compulsory here but schools are not generally obliged to accept any student who wants to go there, even if they live across the street, and some parents have experienced serious difficulties finding a “public” school for their child.
This model exists thanks to a constitutional provision that “the State shall provide *for *free primary education” (my emphasis). So legally, as long as it ensures those schools are there, it’s off the hook.
I’m sure this is not exactly what you had in mind when you asked the question, but still, it’s a very different model to countries where the schools are actually run by the state itself.
*The government has recently got into the business of actually providing schools, but there are only a handful of them.