As I mentioned in my previous post, Hong Kong had an unusual monetary system before returning to the currency board system in 1983. It was (retrospectively, perhaps) dubbed the "free issue" system and that seems as good a placeholder as any, though I welcome suggestions for a different name.
"Free issue" systems are those in which bank liabilities are not convertible into a commodity or foreign currency at a set rate, no party external to the commercial banking system such as a central bank issues a monetary base into which bank liabilities are convertible at a set rate (usually 1:1), and legal requirements compel people to use the local currency. The last characteristic sets them apart from all free banking systems, historical or imagined.
I have found only two historical cases of free issue systems: Hong Kong from 1974 to 1983 and Canada from 1914 to 1926 and again from 1929 to 1935. There are few discussions extant of free issue systems, and those I've seen are fragmentary. John Greenwood's book cited in my previous post is the only place I have seen a balance sheet treatment.
Here is how the Hong Kong system worked. Until 1972, the pound sterling was the anchor currency for the Hong Kong dollar at HK$16 = £1. Hong Kong's three note-issuing banks were in effect agents of Hong Kong's currency board, which was called the Exchange Fund. To issue notes, the banks had to give the Exchange Fund specified sterling assets, such as British Treasury bills, equal to the notes they wanted to issue. (If the banks took notes out of circulation, the Exchange Fund gave them back sterling assets of equal value.)
Hong Kong abandoned the pound sterling as the anchor currency during the upheaval that marked the breakup of the Bretton Woods monetary system. At first it switched to the U.S. dollar as the anchor, but the dollar too had problems, so in 1974 Hong Kong's government floated the exchange rate. The Exchange Fund then began accepting Hong Kong dollar-denominated assets as backing for note issues alongside foreign assets. It converted much but not all of the Hong Kong dollar assets it received into foreign currency.
You can see the potential problem. The note-issuing banks could not influence the creation of British Treasury bills, but they could influence the creation of Hong Kong dollar assets. Buy a security issued by, say, China Light & Power (the local electric company), take it to the Exchange Fund as backing for issuing notes, use the new notes to buy another China Light & Power security, take it to the Exchange Fund as backing for issuing more notes, etc. In principle the monetary base could have become infinitely large. In practice it did not, mainly I suspect because hardly anybody understood how the system worked. It was also in the interest of the note-issuing banks not to barbecue their mortgage loans and other Hong Kong dollar-denominated financial assets with a hyperinflation.
(My example above is hypothetical. Until 1992 the Exchange Fund did not publish a financial statement and the government of Hong Kong rarely made public any information about it. We have rudimentary retrospective balance sheet information, but it would be worthwhile for somebody in Hong Kong or visiting Hong Kong to sift through the archives and write a paper describing the details of how the Exchange Fund worked during the free issue period.)
An essential element of a free issue system is some kind of requirement that people use the local currency, such as a forced-tender law. In a competitive currency system, an individual issuer issuing a currency with a floating exchange rate may create hyperinflation in its own currency, but people have the freedom to use other currencies instead. Damage to the economy will therefore be limited because few people will want to use a hyperinflating currency.
ADDENDUM: A commenter brings up the case of New Zealand, contemporaneous with that of Canada. Both countries later established central banks at about the same time as well.