Money
Until people start gaming whatever de facto monetary system arises to replace the inefficiencies of bartering, but there really are no easy answers here, and much bigger problems to worry about...
People will no doubt game money but as long as their are competing currencies I don't really see a real issue.
Poverty
I'm not saying I'm a fan of welfare, especially not the kind you see in the US, but there is a reason all 1st world nations have SOME form of support for the impoverished.
I suppose if your society is rich enough, there will still be enough charity to make sure no one starves so that's not a concern (unless you have a recession).
But in a society with no law enforcement and no laws, what will stop someone who can't make an honest living from turning to crime? Do we just kick them out once they lose their job/their business fails? Maybe that would be preferable to welfare, but then who makes that decision? Who enforces it? Do you kick out their family too? As an enlightened society, do we allow homeless people, as long as we can somehow make sure they're not breaking any laws?
The reason all 1st world nations have welfare is because they think that is the best way to handle poverty even though it isn't. Poverty is a large topic.
For starters, define poverty. What is poverty to one person is opulence to another. And who's to say people shouldn't live in poverty? Who's to say people shouldn't live in wealth? People should be free to live by whatever measure they are given in life. Being poor could be better than being wealthy. It's all perspective.
But let's say poverty is bad. Giving somebody a fish is much worse than teaching somebody to fish. This somewhat ties into education again. Since welfare gives somebody a fish it is clearly a broken model. Poverty is diminished by people educating themselves and adopting the habits and principles of those wealthier than themselves.
History of America shows that this was clearly the case with the black population as they assimilated into the American culture. Before welfare and all the other government BS they were actually rising out of poverty and poverty was being diminished. When the US dollar actually had value people were able to buy a lot of things for very little money. I really think poverty would be a small concern in an area where so much wealth and innovation would reign supreme.
But if there was poverty so be it. People could help them as they saw fit if they wanted or people could leave them to be if they wanted. There most certainly wouldn't be anyone forcing another person to do something about poverty.
Taxes
Like thelorax said, who builds the infrastructure? I'm not a fan of taxes, but how do you manage a nation-state of any significant scale without east some nominal collection for infrastructure? Is every road going to be a toll road? Monetized through ads (lulz)? How does that work?
Even when you're living in a private condo you have to pay fees. If you don't pay...they kick you out. I bitch about condo fees all the time, but honestly I couldn't suggest a better system - no one is fixing the elevator or taking out the garbage or fixing broken common areas out of the goodness of their heart. You could force ppl to take turns, but then its just another form of tax - a much less efficient one.
And without a reserve fund, what do you do if something catastrophic happens...just hope everyone has enough money set aside to fix it? If a 1000 person building can't function smoothly without some sort of enforced contribution, how would a nation?
Yes I agree. There wouldn't be taxes in the sense of forcing anyone to pay taxes. There would, however, be costs for things. People would pay for stuff. And hopefully, that should would be things people need. The city owners would devise whichever creative and efficient methods they could to pay for the costs of the things people used. Since people would use things on an individual level and not a collective level they really wouldn't care. We have no problem paying for things we want. It's when we pay for things we don't want that somebody else does. That's when we get pissed.
Things like roads could be a toll fee that is done digitally through GPS in your vehicle or something like this. I'm sure road owners would come up with a good way to pay for their roads while keeping customers happy.
Leadership
What would stop someone powerful from seizing leadership eventually, whether through force, money or shrewd political maneuvering?
I think this could happen and this would be the trickiest thing to prevent... especially when we have all these other non-ancap societies that allow these kind of people to develop in. I think a reasonable answer would be to have 2 or more competing locations where people could flee to. If there was an ancap society in Africa and an ancap society in South America competing to be the beacon of freedom then I think it would be a lot harder for a group of people to collude and take over an area. Obviously it's possible, but it's also not profitable. So it really depends on how they would go about it and I think the only safeguard against that would be as many other Ancap societies as possible to minimize the ability for monopoly. It's hard because land is only so limited and we are in that transition phase. Once we got enough people to open their area up to Anarcho-Capitalism I think there would be a tipping point where it would no longer be possible for mafias to establish. Sure, they could on a small scale, but with enough Ancap societies, maybe 10% or 20% of the world, there would come a point where it would be too expensive and there would be too many alternatives for people to go for the mafia to establish themselves as a global threat.
My thoughts anyway, obviously there is a lot of discussion to be had on these matters. Thanks for the great points.