The Fasori Gimnazium in Budapest, while it was open between 1864 and 1952, might fairly be claimed to be the best high school in the world. It educated Eugene Wigner, John von Neumann, Edward Teller, Alfred Haar, and John Harsanyi.
So it might be useful to know what they were doing right.
Laszlo Ratz, who designed the curriculum, was the driving force behind the school. He founded the high school math journal KoMaL, which presented challenging problems so students could write in solutions. The journal is still in print; you can see sample problems here. Harsanyi and Erdos, along with other prominent mathematicians, were especially good at these competitions.
Ratz also cultivated personal relationships with his most talented students, inviting them to his house and giving them book recommendations.
Here is some biographical information about Ratz, which gives some insight into his ideas on curriculum.
The basic principle of the Fasori Gimnazium was that students were presented with examples first, and rules for how to solve the problem only after they’d tried to figure it out for themselves. They also practiced with real statistics, from things like national railway schedules and tables of wheat production.
Ratz had a particular axe to grind about calculus: he insisted that the concept of derivatives be taught by starting with finite differences.
Wigner’s recollections about high school noted that they learned Latin, poetry, German, French, botany and zoology, and physics from a history-of-science perspective. The physics teacher had written his own textbook. He remembered Ratz as exceptionally friendly and encouraging, giving private lessons to von Neumann and lots of books to himself.
It’s hard to determine which, if any, of these things made the Fasori Gimnazium special. But it does point in some useful directions. One-on-one attention from exceptional teachers, a focus on problem-solving and examples, math contests. It matches my intuition that you only really understand a mathematical concept when you’ve computed it by hand with examples.