We study a continuous time Bayesian quickest detection problem in which observation times are a scarce resource. The agent, limited to making a finite number of discrete observations, must adaptively decide his observation strategy to minimize detection delay and the probability of false alarm. Under two different models of observation rights, we establish the existence of optimal strategies, and formulate an algorithmic approach to the problem via jump operators. We describe algorithms for these problems, and illustrate them with some numerical results. As the number of observation rights tends to infinity, we also show convergence to the classical continuous observation problem of Shiryaev.