An Iowa-based philanthropist and architecture aficionado has offered a $300 million reward to any city anywhere in the world that dares to hire someone other than Frank Gehry to design its gleaming new art museum.

"Don't get me wrong, I like iconoclastic, swoopy structures that look like bashed-in sardine cans as much as the next guy," says the philanthropist, who wishes to remain nameless for fear of enraging close friends in the art world. "I like Czech dance halls that look...