PITTSBURGH—Tim Cook is fighting the sky-high cost of a college education by constructing his own school here without expensive buildings or well-paid deans. Classes are taught in local coffee shops. The administrative staff of two works in a church basement.

The Saxifrage School, Mr. Cook's two-year old experiment, is seeking to upend the traditional notion that college students need a sequestered, ivy-covered campus—and will endure the price tag that comes with it. He is gambling that for a nominal tuition—$395 a class—they...