Which aspect of the macroeconomy does real business cycle theory find most difficult to explain?
I would say that the apparent success of Keynesian stabilization policy in the United States since 1945 might be the most difficult aspect to explain. In my view, the pace of economic change has increased in recent years, and that should cause people to make more mistakes with their investments in human and physical capital, leading to more frequent and more serious recessions in the RBC world. In fact, however, postwar recessions have been relatively mild, and we've had some very long periods, notably 1991-2000, without a recession.
One possible explanation for this that might salvage RBC is that we have what I call an elastic economy. Otherwise, you have to give some credit to Keynesian stabilization policy. In fact, one could say that the most dramatic Keynesian intervention ever was the fiscal policy of the current President, which arguably kept the collapse of the Internet bubble from turning into a re-run of the 1930's.
Perhaps the topic of the macroeconomics of the Internet bubble and its aftermath would be my choice for Tyler's "write your own exam question" question. But I'm not sure how I would answer it.