Defense is probably an obvious example. There's no way that a private company or any other non-governmental group could ever possibly provide the services of national defense that the United States government currently provides.
EDIT: On the topic of health care, government health care programs in various European countries provide medical care at a substantially lower per capita cost than does the US system of private insurance companies. I'm not sure what you mean by "efficiency", but they are cheaper and don't leave 15% of the population uninsured, so that's a plus.
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