Tyler [Cowen] once walked into class the day of the final exam and said, “Here is the exam. Write your own questions. Write your own answers. Harder questions and better answers get more points.” Then he walked out.Tyler goes on to say later in the quote that the concept would probably work better for one question than an entire exam. I can see this type of question working better for specialized university courses than for standardized exams.
For the exams I use for my course at the GGS, I routinely write one question that I divulge the week before the exam, partly to encourage them to come to the review class, and partly because the question really does require a fair amount of thought and research. When I taught an art song class at UBC, I once wrote an entire final exam consisting of broad questions and opinion pieces that covered the entire year's work. The students were welcome to use all their notes from the class. As an open book exam, the students were highly motivated to assemble a large body of notes to take to the exam, which they would then have on hand to consult as they entered the profession.
One of my teachers described a similar idea in an exam they had taken in the 1970's - the final exam required the students to describe in detail what they had learned throughout the course.