None. All states and U.S. territories have some mechanism for homeschooling. Some states’ requirements are quite burdensome, while in the lovely State of Oklahoma, the State Constitution gives parents complete freedom in managing their children’s education.
I call them "free states" vs. "slave states." "Free states" (like OK, MO, etc.) don't require any state permission or notification to homeschool.
Also, to the poster who mentioned the 1925 Pierce decision - Pierce codified the right to send one's child to private school - but also in Pierce, the court established that the state did have the right to regulate private education.
In MO, there was a federal court ruling called Ellis vs. O'Hara, which said that the old state law provision requiring homeschools to be "substantially equivalent" to public schools was unconstitutional under the Pierce decision. Unfortunately, most of the states that still have "substantial equivalency" in their state laws are not in the jurisdiction of that federal court, and to my knowledge not even HSLDA has filed a federal suit to remedy that.