Australia, like most countries, faces high and rapidly-rising drug costs. There are longstanding concerns about pharmaceutical companies inappropriately extending their monopoly position by "evergreening" blockbuster drugs, through misuse of the patent system. There is, however, very little empirical information about this behaviour. We fill the gap by analysing all of the patents associated with 15 of the costliest drugs in Australia over the last 20 years. Specifically, we search the patent register to identify all the granted patents that cover the active pharmaceutical ingredient of the high-cost drugs. Then, we classify the patents by type, and identify their owners. We find a mean of 49 patents associated with each drug. Three-quarters of these patents are owned by companies other than the drug's originator. Surprisingly, the majority of all patents are owned by companies that do not have a record of developing top-selling drugs. Our findings show that a multitude of players seek monopoly control over innovations to blockbuster drugs. Consequently, attempts to control drug costs by mitigating misuse of the patent system are likely to miss the mark if they focus only on the patenting activities of originators.