I think most people know that Somalia has what is called a "failed state" in some quarters, and a "stateless society" in others. They have no central government, and yet they have police, laws, and courts (as they know them; not as we know them). They also have a national currency .... without a nation.
The Somali shilling is left over from the old government. I forget whether it was socialist, democratic, or a dictatorship. Doesn't really matter. A central government can issue currency and enforce its value by changing the amount of currency in a controlled manner. Typically, this is done by shooting counterfeiters. Counterfeiting is a very serious crime in every country.
And yet, since there is no functional central government, how can the Somali shilling possibly work? You would think that counterfeiters would simply print up more and more and more bills, since there's nobody to stop them. But that's not what happens. Yes, new bills have been printed up in the intervening years by private parties. But the currency has not suffered from unlimited inflation, and I think I know why.
A fiat currency functions because the government controls the supply of currency. Yet it also controls the denominations of the currency as well. The government can inflate the currency as much as it wants through the expediency of printing up a new bill with extra zeros, and paying its debts with these bills. By declaring that the national currency is legal tender, everyone must accept it as the equivalent of ten of the next smallest denomination.
However, that's not what happens in Somalia. There, the denominations of the bills is fixed. There is no central government to pay people with new denominations, or to force legal tender laws. Nobody would accept a new bill because "it's not money". Let's trace the state of currency affairs.
At the end of life of the former central government, there was some supply of bills. Two things will happen over time. First, the economy will grow, because any economy in which people trade is an economy that will grow. Second, the existing bills will wear out, get lost, be removed from circulation, or otherwise become unavailable. Both of these effects will increase the demand for currency. For any item of trade with a fixed supply, an increased demand will raise the price of the item, and that includes money.
So, over time, the shilling becomes more valuable. Against that effect you have the counterfeiter, printing up new bills. Initially, he will print up lots of bills, and make a windfall profit. However, as thse bills make their way into circulation, the value of the largest denomination will fall until its value matches the cost of printing. No counterfeiter will bother to counterfeit unless they can make a profit, so they will stop printing up new bills.
What makes this system different is that a government would react by adding zeroes to the currency, e.g. Weimar Germany in the 1920's, or Turkey in the early 2000's, or Zimbabwe as I write this. No new denominations can be created, so the value of the currency stays stable at about the value of printing the highest denomination. As soon as its value increases, counterfeiters will print more. Competition between counterfeiters will reduce their profit to an amount approximating their costs, so the harm from counterfeiting will be minimized. Society will have a stable non-fiat paper currency.
This answer generates questions (as do all good answers). First, that the currency will tend towards a single denomination of bill (presuming that all denominations cost equal amounts to print). Second, it suggests that the value of the largest denomination will be fairly low, so that people will have to pay with lots of bills. Third, it may be that people will rip the largest denomination of bills in half or in quarters. Fourth, people may staple ten of the largest denominations together to make a single "10" bill. I don't know if Somalis are doing this now.