It is widely known that the market for health care services is rife with several market failures. See this seminal paper by Kenneth Arrow. It is also widely accepted that the United States, with its private health insurance market, is the best representation of these failures.
The Economist has a graphic that traces the major sources of wastage and inefficiencies and an estimate of the amounts. As the article points out, the pay-per-service model of insurance creates perverse incentives - "the more services a hospital provides, the more it is paid".
Ezra Klein makes the interesting point that even in comparison to single-payer systems like Canada (government is the only insurer) and UK (the government is not only the sole insurer of note but also employs most of the doctors and nurses and runs most of the hospitals), US, with its private insurance dominated health care market, has a much larger government health care system. It also has the highest private health care spending.
Health care, especially in light of the US experience, is also a classic example of "how it can be possible for unregulated free-market health-care systems to cost more and deliver inferior care than strongly regulated systems with heavy government involvement"!
See excellent graphics from Kaiser Foundation comparing health care costs across developed economies.