Health insurance and fire insurance
The Great Fire of 1666 started in a baker's shop on Pudding Lane, consumed about two square miles (5 km²) of the city, leaving tens of thousands homeless. Prior to this fire, London had no organized fire protection system. Afterwards, insurance companies formed private fire brigades to protect their clients’ property. Insurance brigades would only fight fires at buildings the company insured. These buildings were identified by fire insurance marks.
The parallel is striking: those who could afford fire insurance would subscribe to it, and sometimes would pay more than one insurance company because (1) you never knew which brigade would be faster to arrive at your house, and (2) if the first brigade to arrive didn't see their company's sigil on your building, they'd actually interfere with the arrival of your insurance company's brigade!
And those who couldn't afford insurance would stand by and watch their houses burn down.
Until government decided that an effective firefighting system was a necessity for society, and took over that responsibility.
I'm not arguing in favor of a particular form of government intervention in the health care crisis. The current mess is a result of the 1960s tax code and it will take some time to unwind it. My point is simply this:
The private market cannot be effective at providing anything universal, and therefore if we as a society decide that something must be available to everyone, government must inevitably be involved, either as a provider, as a guiding hand on a licensed monopoly, or through mandates.