The problem with rent-seeking is that I don't really know what to do about it.
Rent seeking seems to happen wherever there is some scarcity or monopoly (natural or artificial). E.g. patents (artificial), real estate, infrastructure (both natural).
In the case of infrastructure, the answer is relatively simple - nationalize it (or strictly regulate it), and offer it to multiple operators at fair (and equal) prices. This is what is happening already in e.g. Europe (for electricity and mobile networks). It can even work for other scenarios - for example, when the Greeks were forced to "privatize" their airports, they didn't actually privatize them, they just rented them out for a number of years - that way, whenever the lease is up for renewal, there is still a market happening, so it's more competitive than ownership/monopoly.
For patents/copyright, it's considerably more complex; I think we can all agree that current regimes in e.g. copyright law and software patents feature way too long monopolies, but it's not so clear if that's the case in other fields, especially farmacy.
For real estate, it's hard to come up with any solutions. As long as there's a benefit to living close to other successful people, rents/prices in high-prodoctivity areas will keep on rising. Taxes could help (e.g. by making buy-to-let less profitable), but they won't solve the underlying issue. I'm hoping that eventually technology will improve to such an extent that there will be no advantage in living in the cities (i.e. remote working/communications technology or much faster transport).
Are there any other worrying examples of rent-seeking that you had in mind?