In an Anarcho-capitalist society, where there is no government, who would build the roads?
The roads might be built and maintained by different persons: private capital investment, community ownership (an especially good option for a transition between our mixed economy and a free market economy), public-private arrangement, monopolies, or possibly, no one.
The roads could be paid for without a centralized tax system by using a toll road system. Toll roads were popular in the 1800's and often financed by individuals without any impetus from the government, thus some roads would surely be built or maintained without any intervention. Many corporations now work in the field of toll collection processing for governments under the Build Operate Transfer system that also removes the need for a government to operate the road. Ideally, the roads would be built by many private investors to the extent possible, but some could be owned by the public and contracts given for work to maintain them, but corruption in contracts would be closely monitored.
A rare possibility is that all the roads would be purchased by private investors immediately. Drawing a parallel to the railroads of the 1800's, it is possible that since the infrastructure already exists, (which lowers "entry"/initial investment costs making it a better investment) businessmen might attempt to monopolize the road system. This would remove concerns about the upkeep of an interstate highway system for trade and national defense, but may raise other concerns that may be difficult for such a society to address, such as further horizontal integration. If this happened, one person or company would own all the roads and probably related industries, such as trucking companies.
The caveat to this is since there is no government, there will be no law nor permanent policy passed regarding building and maintaining the roads, specifically the interstate highway system. So at some point if relationships become strained between communities or there are other priorities, there is the risk that some or all of the interstate highway system would stop being built and no one will be building the roads the nation needs.
I think the marked answer is a correct and good one. Direct to the question, but I also feel the question itself has a built in assumption. That is that roads are the majority and necessary choice for transportation. The question, "who will build transportation system" is more applicable.
Roads are actually a burden not only on taxpayers but on anyone that has to buy and maintain a car. Its a big hurdle people have to get over to 'travel freely'.
Alternate transportation system would most likely arise to meet the needs of those who do not want to pay the high cost of car ownership or simple do not want to own a car.
Most people do not ponder why it is so difficult to get train system or alternate transportation system to work easy. And why must bus & streetcar systems be so heavily subsidized? It is because government has created a transportation monopoly. This monopoly basically dictates that roads are the only way to travel freely. It kills off other options. It definitely kills off free market options as no one can compete with 'free' money from taxes.
So the answer really is that people will put money into their desired mode of transportation and those will develop based on resources given to them. Thats what free markets are great at doing. Voluntary diplomacy in resource control.
First ask what the government is. The majority of governments emerged as either landlords (King Arthur) or port monopolies (US government tariffs). They collected "tolls" or "fees" and later spread their land or infrastructure monopolies to more areas. Taxes are fundamentally a road toll, since the majority of arrests are made at traffic stops which is where any tax evaders would be caught. The other reason is if you refuse to leave a tax liened house and the gov shuts off utilities, in which case the government is a utility monopoly.
So the government is already a private entity and these questions are meaningless.