We can probably look to historical examples for a plausible answer. So your reference to US coinage isn't too far off the mark. However, you're a little biased... within your own lifetime, it's certainly only that list. But in the last few centuries there have been additional denominations:
- The half-penny
- The two-cent
Also, there are bullion coins (made of gold, silver, and recently, of platinum!)
- The \$5
Historically, I'm only aware of \$10 and \$20 coins though. So, we're probably talking about ten denominations that saw actual use. And not all were available simultaneously either, with as many as 7 or 8 at any given moment maybe being minted (though with circulation times being what they were in the 19th century, no doubt all ten were being used somewhere).
As a counter-example, let's use pre-decimal British coinage, because they were a little bit crazy with their stuff.
- 1/4 farthing
- 1/3 farthing
- 1/2 farthing
- Full farthing
- Half penny
- 3 farthing
- Full penny
- 3 Half-pence (1.5 pennies)
- Half-groat (don't even know what this is, but wikipedia lists it)
- 2 penny
- 3 penny
- 6 penny
- Quarter angel (gold?)
- 2 shilling
- Half crown
- Half angel
- Half sovereign
- Half pound
- Double crown
- Half guinea
- 2 pounds
- 2 guineas
- 5 pounds
- 5 guineas
Again, not all of these were being minted simultaneously. But I've also left out quite a few that were minted only for a few years at a time... these are the list that were minted for about a century at minimum. This is approximately 30 denominations.
Part of the reason there are so many is because this lists includes denoms that are pre-paper-money. (If we included paper money, we'd easily double the denoms for the US and most other currencies.) That's how it has amounts like the 5-pound coin (a minor fortune, especially if you're a Dickensian orphan asking for more soup).
So, anything from about 8 denominations on up to 30-40 isn't without precedent, and maybe even with most of those circulating or being minted simultaneously.
Now that we've covered that, we should really discuss just what is needed. Because if an economy needs another denomination, and if the country's planners are even slightly competent they will see the need and try to respond, no?
Coins will serve (in a medieval environment) the same purpose as paper money does for us today (or more properly, did serve us until the late 1990s). So on the high end, denominations need to be large enough for the largest transactions commonly occurring without using too many of them. In modern times, it'd be "can we fit enough $100s in the briefcase to buy our cocaine shipment", but in medieval times it would be "can this purse fit enough currency to let me smuggle in my sailing ship's worth of brandy" or something like that. Coins are small, and large denominations are going to be precious metal.
Exceptions: For the very largest transactions, like kings buying and selling their kingdoms to the Medicis to finance wars, transactions will occur in the same manner that these things occur today. A bank will put the amounts in a ledger, and keep track of the interest to the fraction of the smallest coin amount. You didn't pay for your house with a suitcase full of cash, and neither did I. So there's no need for the $10 billion florin coin.
On the low end, how granular are your transactions? The British had fractional farthings because the poor wouldn't be found with pound-coins (if they were, they were likely about to be shipped off to Australia), but the poor still tend to buy bread and what-not. They might be paid just a few farthings per month, but needed to make more than a few transactions per month (this is pre-refrigerator, pre-preservative food).
Between these two extremes, we can postulate that increments only occur every so often. If the two extremes are close, then there might only be a few intermediate values, but if they are extreme (as was the case in the British Empire), then there will be many intermediate values. And with less competent economic management, not all of these values will have available denominations, but the more competent it becomes, the more of those values will see actual coins. I really do like the Brits as an example of the most extreme you'll likely see.