Minarchism is a political philosophy advocating the reduction of government functions to a minimum scope, providing people with only bare-bones services such as law and security. It's considered a form of libertarianism, and since it finds some useful role for the state, it's incompatible with anarcho-capitalism, another major form of libertarianism. Ayn Rand's views could be described as minarchist.
Minarchists advocate for a "night-watchman state" that is not responsible for the education, health care, employment or transportation of its citizens, neither makes it any use of natural resources in its territory. All of this is instead held privately or publicly, but is never susceptible to any interference of the state, its law or its representatives. Minarchy is, of course, different from anarchy, since the latter term means a complete absence of a government with all services, including even law and security, done or exercised by people themselves.
Even the Austrian Economists do not advocate for this. The almighty Free market fails utterly when dealing with Common Goods, that is, goods that are rivalrous (we can't both have the same one) and non-excludable (you can't stop me from taking it). For example, there are only so many fish in a location, and Alice can't stop Bob from fishing there, so Alice and Bob catch too many fish and next year there are practically no fish left. Alice and Bob could make some sort of agreement, but nothing prevents Charlie, Denise and Emily from still fishing. So everyone is poorer than had there been some sort of limit on overfishing.
Next comes Externalities, which are detriments (or benefits) that the producer of the goods doesn't have to deal with. For example, pollution. If a factory can make significantly more money by just dumping their waste in the local river, so long as the owners don't drink from said river, it probably will. This generally harms the local population far more than the company gains from not properly disposing its waste, so overall it's a net loss for society.
Things like education, safety regulations, roads, research subsidies, and so forth often add to the economy more than they cost. Welfare and subsidized housing are often far more effective at reducing crime -for cheaper- than extra police alone. While not every government program is cost-effective, most are. What Minarchists have a hard time understanding is that their income depends on things like roads and communication systems being available, that having access to adequate healthcare increases worker productivity, and a whole slew of other things. Even if you built your business out of gumption and bootstraps alone, virtually all of your customers and employees relied on something provided via government, and without those customers you wouldn't have that business to be taxed in the first place. It's far better to pay 50% tax on $200,000 than pay 5% on $20,000.