What makes big, course-spanning questions so great is not that they motivate students with a tantalizing question. No question that I ask is going to be able to motivate students 5 months after I ask it. Any moment of curiosity will have passed. For motivation and engagement, we need daily questions, curiosities and (for +10 Meyer points!) perplexities.
I think that the big unit/course spanning questions are wonderful because they provide meaning to the curriculum. It's harder to ask the question "Why are we learning this?" when what we're learning is clearly situated in a larger, obviously meaningful framework. For instance, the question "Is there life on other planets?" naturally leads to the questions "What are the conditions for life?", "How hot are other stars?", "How far away are planets from stars?" and "What's in the atmospheres of alien planets?" Bam, there's your calculus-based intro Astronomy course. And while students in a different class might wonder, "What good are absorption lines?" my bet is that students in this class will (a) be more likely to situate them correctly as helpful in determining temperatures of distant stars or atmospheric content of exoplanets and (b) won't think that astronomy is useless and boring. So that's what we're going for here, I think.
And now, a problem. Let's partition the world of course-spanning questions into the purely mathematical and applied mathematical questions. Let's take an applied mathematical question such as "Can we predict the motion of a basketball?" or "How do electronics work?" or "Can we beat the stock market?" If we really and honestly pursue these questions, we're going to have to go beyond our mathematics, since we're going to need to use the tools of physics, or economics, or engineering. In other words, doggedly pursuing non-mathematical questions quickly leads us out of the mathematical domain.
On the other hand, rich mathematical questions don't typically do the work of being obviously meaningful to students. The best that I can think of is "What's a number?" which I imagine as a narrative arc spanning the first bits of a first year of Algebra.
This is a long-winded way of saying that I think we're either looking for mathematical questions that are big and basic enough to motivate this month-long investigation, or applied mathematical questions that are closed under honest inquiry.