"... I can't really think of anything ..."
No real surprise there ... but that means little since there are indeed private companies that either presently do - or would wish to - compete with government-provided services. Policemen/firemen may not be good examples, but take school busing. It used to be provided by private companies in many areas of the country, but now those services are provided almost exclusively by government-funded facilities - and the government has such a tax advantage that no one really believes a private firm can be cost-competitive with such a "free" (yeah, right) service.
The fact you can think of none most likely means you're not trying too hard since it doesn't fit your agenda of opposing the FairTax.
Bigun commented on this subject a number of threads ago, but you must not have been paying attention. If the government decided in all its Congressional wisdom to take up the manufacture and sale of fuel-efficient automobiles, for example, do you think that many existing private companies would be cost-competitive for very long while being burdened by a tax load that the government did not have to pay???
But in the FairTax world, its the private company that doesn't pay the tax while the government enterprise does. This makes more sense to you?
The current system has both private and government employees paying income tax. The one area where government might have an advantage is not needing to make a profit for investors. That, however is not a tax issue.
Who wants to be in the business of busing children to school? The government is not in the business of competing with private enterprise and in the few cases it does is because the people demanded the government provide those services.