Examples for me are the criminal justice system and to a lesser extent, venues for civil justice. As to the criminal justice system however, I think if we think about this from the standpoint of evolving the U.S. into a minarchist government, there would need to be an amendment that dictated that only actions involving a victim could be enforced criminally and that in all cases, the specific victims must be indicated on the complaint and that the victim would have the option of waiving the prosecution upon payment of restitution with one exception (that I can think of) and that being a case where the victim is being intimidated which would be a severe punishment crime. This exception would be in place to prevent thugs from committing crime with impunity.
The civil justice system is even now largely privatized -- most civil cases never reach the courts, being settled before they get there. People enter into agreements where they agree to arbitration all the time and waive rights afforded to them by law all the time. There is room for improvement and more privatization, but if it remained largely unchanged, I don't see a big problem with it.
I think the military has a role too and I haven't seen a workable solution to replacing the military as we know it. I don't buy into the idea that we could simply rely on individuals to protect themselves one by one or as small bands of militias against invading armies. I think the answer lies in some sort of decentralized control structure. With the speed with which attacks can be made in today's world, and that speed increasing all the time, I'm not sure how decentralization would work, but I think at least a partial solution lies in this approach. The fact is, there are countries that don't subscribe to anarcho-capitalism and they have big weapons paid for by their own non NAP practices that must be defended against. Voluntary funding might be part of the solution. Complete transparency vs. military secrets could be another part of the solution. And the biggest problem isn't the stealing of money to fund this nominally important function, but the potential for it to be used against those it is supposed to be protecting, or used against foreigners who have not aggressed.
I'm also of the opinion that government roads are so essential to the promotion of free trade and free association, and because of the potential for privatization to interfere with that, that the building and maintenance of government roads is a proper role of government just from a pragmatic point of view.
We have systems in place now that have private contractors bidding for the building and maintaining of roads under supervision of government entities like CalTrans in California. I see that as largely successful. I do think traffic tickets are largely handled wrongly and this is where my "where's the victim" rules could come into play.
Registration should be outlawed, not just of vehicles, but period. I don't want the government keeping records on individuals like a social security number, an FBI file, a DMV file, etc. and I can't think of any good reason to maintain records that identify individuals. If a crime is committed with a victim, obviously a file would need to be maintained during the investigation until all matters are resolved that pertain to it. When the case is closed and penalties satisfied, the file should go to the perpetrators and the victims and not maintained by the government. I see more problems than benefits from such record keeping and of course one has to violate the NAP just to maintain archived records.
I'm hoping I was coherent here. Lots of other things on my mind today.