I think the question is a good one because many people believe that, no matter how bad the existing government may be, a failure to establish a government would be worse, with rogue gangs running around, and foreign nations possibly threatening to conquer the land. If we could show them more precisely how a voluntary civil authority could fill the gaps where they see possible market failure, then we can win the intellectual argument. That argument has not yet been won, despite the work of Murray Rothbard and other voluntaryists, who neglect to answer the fundamental questions as to how justice would be established, how a country could defend itself in a war, and how several sundry types of order, which the government now provides, would be replaced, such as fishing rights, air traffic control, etc.
So, the broader question is, would there be some remnant of a government in a free society? I think there would be, simply because nothing would stop it from forming under the non-aggression principle. Speculating farther, following are the types of things in which it might be legitimately engaged. I believe that a principled minarchy can organize and pass civil guidelines (in lieu of laws), which might be followed due to the size of the organization, as it represents the will of a large group of people. This would give society guidance in many areas where proper action is not clear. Some of these areas are as follows:
1) Establish a database of records, so that people can voluntarily record their birth date and location.
2) Record land surveys and assessments to be used in private courts.
3) Establish guidelines for ownership of airways to benefit air traffic control, which could be implemented through private means, by the airlines. Courts could then have a guideline for determining when a rogue pilot may have violated airspace recognized as owned by an airport, or air traffic control network, for instance.
4) Establish a means by which ownership of new property can be claimed and delineated, as in Earth orbits, broadcast airways, the moon, and fishing rights.
5) Raise an army in times of war, providing they can raise the money for it.
6) Intercede in resolving disputes beween parties who cannot otherwise agree, on behalf of one of the parties. For instance, say Party A is robbed, and has his defensive agency drag Party B into a jail without trial. The principled minarchy could hire someone to intercede on behalf of Party B, if no one else will, and allow him to be tried and convicted in a court it deems to be fair.
Thoughts?