If there's a state, it's natural to try to corrupt it to use its political power. If there is no state or that option looks unappealing, the alternative is to get some political power of the sort that grows out of the barrel of a gun, i.e. to be organized crime.
The implication of the question, that the state is necessary to produce a monopoly, fails to give the full picture. It's not as much that the state is necessary, that economic power can eventually be translated into enough force to subvert the rules. In the absence of a state, that force just takes the shape of an authoritarian proto-state, itself.
https://mattstoller.substack.com/p/this-is-not-a-democracy-i...
[EDIT] OK that was a bit glib. My point is things are a lot more complicated than "capitalism does this, government does that" and the two are essentially inseparable. Further this entire thread is full of "markets = capitalism" which is missing the mark—tendencies toward free market policies are a defining feature of capitalism, but free markets per se aren't identical to capitalism.
Governments aren't fundamentally needed per se to create a free market. But rules are needed. And whatever system we have to enforce those rules would resemble a defacto government. How that governments power is distributed, who's interests it represents, and how far it goes in regulating the market may vary.
Without someone enforcing rules to keep monopolies in check, I think those monopolies would become defacto governments. And they wouldn't be democratic. This is what I mean (but did not carefully express) when I said that some government intervention in the markets are needed to prevent the powerful from take advantage of the weak. (And it can go the other way too... governments tilting the scale in favor of existing powerful interests.)
Would that be different from the way that governments normally maintain a regulatory environment for businesses to expand?
Can free market capitalism even exist without the property enforcement that what we'd consider as organized crime or governments provide?
My second point would be: monopoly or not, it's not clear that they would have maintained the advantage they had. Of course we know that the government intervened, and some people will say that's what broke MS's alleged monopoly, whlile others will say it made no difference, and still others will say that they couldn't have maintained it for long in a free market anyway. We'll never know, since it happened the way it happened, but personally I don't see how the government's actions made the success of Linux, Apple and Google products possible where they weren't before; therefore I find the last argument more convincing - that is, that Microsoft had the jump on everyone, but it was never going to last. A free market doesn't guarantee that there won't be a clear front runner in any given competition.
People who usually ask "give me an example of a monopoly not maintained by the government" are usually not asking in good faith. Many laissez faire capitalists explain away natural monopolies, or pretend they don't exist. So perhaps you expected this answer, and are already prepared to dismiss it.