I kind of agree with you on anarcho-communism, and I don’t think it has ever happened, before, in history, but it would still be possible if the entire community agrees to work for each other; the problem with that, obviously, would be that not everyone can be trusted to do that, and either people realize that trading is a better system, or an authority emerges to enforce the labor. Since socialism go in between capitalism and communism, your definition of pure anarchy sounds like my definition of anarcho-socialism. And anarchism-socialism would still require some of the state to enforce an economy that provides certain services.
In anarcho-capitalism, the free market prevents corporations from becoming monopolies, since most monopolies are products of subsidies, which come from socialist/communist government. No entities would exist to prevent other corporations from overthrowing monopolies.