...not just...

Well, the condition I describe in words above doesn't directly limit my ability to bet on any question at any time -- it just specifies one possible bet in one particular set of states of information.

However, the condition can be turned into an integral equation in which the prior density is the unknown quantity. The equation can be explicitly solved to give an analytical expression for the unique prior density which satisfies the verbal description above. Since I posted the question in this thread, you can probably guess the punch-line: the prior is improper. In fact, it's the standard "non-informative" prior for the normal distribution with unknown mean and variance.