The cream of Denmark’s entrepreneurial class is gathering in Skagen this month, as the short summer season kicks into high gear in the resort town at the very top of the country’s mainland.

The vacationers, drawn to the area’s natural beauty as well as to its artistic legacy, live it up at a few local bars and restaurants before hunkering down for the night in their million-dollar homes—surprisingly modest cottages that rarely exceed 2,500 square feet.

“We...