One important category of exceptions comes from the classical economists of the 19th century. Adam Smith wrote "The third and last duty of the [government] is that and erecting or maintaining those public institutions and those public works, which, although they may be in the highest degree advantageous to a great society, are, however, of such a nature, that the profit could not repay the expense to any individual or small number of individuals, and which it therefore cannot be expected that any individual or small number of individuals should erect or maintain." In modern economics these are called "public goods." Smith didn't provide any examples, but John Stuart Mill gave the example of a lighthouse.
According to Mill, it would be impossible to try to charge seamen according to their use or benefit from the lighthouse, and might therefore be most convenient for the government to provide the lighthouse and pay for it out of tax moneys. This is one aspect of a "public good:" for some reason, it is difficult or impossible to charge those who benefit from it. It is easy to see how this would be difficult for charge for the lighthouse service. Try to visualize toll booths for the purpose out in the ocean!
Another thing about the lighthouse is that its cost does not depend on the number of ships that use it. The lighthouse is absolutely indivisible, and the cost of keeping it up is fixed. This is another aspect of a "public good" in general. The cost of a "public good" is independent of the number of people it serves.
This is an extreme case of "economies of scale." We recall that indivisibility causes economies of scale in general, but in this extreme case there is just one possible scale of operation -- one lighthouse -- and so long as it operates it costs the same, no matter how many ships it warns off the rocks.
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