What is the role of the government in a free market economy?

When people say "Free Market" economy, they have in mind what most economists would call a True Free Market Economy. When people refer to a "Market" economy, they generally have in mind the economists' "Mixed Market Economy".

This answer is based on a True Free Market economy. Look at the Related Questions section for answers around a Mixed Market economy.

Besides functions which aren't specifically related to the economic model that a country employs (e.g. defense and policing, social behavior regulation, etc.), here is what a TFM economy requires from the government:

1. A stable single currency to conduct all economic activity in
2. An independent, consistent, and stable legal framework and judicial system to resolve all contractual disputes
3. Protection of individual rights of property
4. Protection of the sanctity of contracts (i.e. not to just resolve contract disputes, but to enforce contracts legally entered into)

That's it. Government in a TFM is very limited in its economic participation, and are essentially a bystander whose role is very much that of the umpire - to resolve issues around rules, and enforce rules. It is NOT responsible for the creation of those rules.