In addition to Alaska, Iowa, Missouri, Nebraska, Nevada and Wyoming, the counties are clustered in southwestern Arizona, western Colorado, southern Mississippi, central North Carolina, as well as parts of Georgia, Virginia and West Virginia.
There are also two counties in California -- Monterey and San Benito -- and two in Florida -- Hardee and Monroe -- with only one choice of insurer and extremely high premiums.
There are many more counties -- including most in the southeastern United States -- where there is only one insurer, though premiums are not quite as high.
Republican congressional leaders argue that eliminating the mandate penalty, which is technically a tax, will save many Americans money, especially coupled with other cuts promised in the tax legislation.
"The individual mandate isn't just any tax. It's a terribly regressive tax that imposes harsh burdens on low- and middle-income taxpayers," Senate Finance Committee Chairman Orrin G. Hatch, R-Utah, said as his panel debated the tax bill this month.
"Zeroing it out means we have a chance to provide greater tax relief to middle-class families, through both reduced penalties and lower overall rates."
The insurance mandate -- which subjects Americans who don't have coverage to an annual penalty of $695 or 2.5 percent of income, whichever is larger -- has always been the most unpopular part of the 2010 health care law, often called Obamacare.
But most experts agree that eliminating the mandate without some other mechanism to encourage young, healthy Americans to get coverage could have disastrous consequences for the individual insurance market, which serves consumers who don't get coverage through an employer or a government plan such as Medicare or Medicaid.
Independent analyses by the American Academy of Actuaries, the Congressional Budget Office and others suggest that without some kind of penalty, many Americans would wait to get insurance until they are sick. That would push up premiums, driving away more consumers and causing what is known as a death spiral.
Several states, including Washington, experienced just this kind of market collapse when they tried guaranteeing their residents coverage without any kind of requirement that all people -- including the young and healthy -- get coverage.