There's no universal law for what denominations of bills must be made, but there are some general guiding principles. I come up with a slightly different list than @InstantMuffin. Here's mine:
It must be possible to come up with a set of bills totaling any desired amount. But that's easy: You have to have a 1.
You should be able to make any likely amount with a manageable number of bills. For example, if all you had were 1's, and you had to pay 653, that would require counting out 653 bills. Or if you had just 1 and 100, 653 would still require 6 hundreds and 53 ones, still a pain to count.
It should be easy to do the arithmetic to add up the total value of a stack of bills. If your denominations were 1, 6, 13, 28, and 87, it would be much easier to make arithmetic mistakes if trying to count in your head than if they were 1, 10, and 100.
The number of different denominations shouldn't be too large, or it gets hard to manage them. Like in the US, cash registers generally have 5 bins, for \$1, \$5, \$10, \$20, and "bigger". If you had 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 15, 20, 25, 30, 35, 40, 45, 50, and 100, this would be a pain for anybody who has to sort money: the number of piles gets too large.
In the U.S., few people use cash for very large transactions any more. So despite inflation, our largest bill today is \$100, while in the past we had \$1,000, \$10,000, and for a time in the 1930s even \$100,000 bills. I believe nothing bigger than \$100 has been printed since 1969 (wouldn't swear to the detail). (I can't imagine wanting to walk around with \$10,000 bills in my pocket. What if I lost it? I remember once when I worked for a non-profit taking a deposit to the bank, I had \$30,000 in my briefcase. And I was thinking, I would really hate to get mugged right now. But anyway.)
So it makes sense to have 1, 10, 100, etc, as high up as makes sense for your currency. It's easy to count, keeps the number of bills you have to carry around manageable, etc. Adding 5, 50, etc also cuts down on the number of bills you need to carry and how many you have to count out for a transaction, without making the arithmetic difficult. Other denominations can be problematic. So you see 2's and 20's. I think any other denomination is rare.
I presume a society that used base 12 would use denominations that are round numbers in base 12, i.e. 1, 10 base 12 (i.e. 12 base 10), 100 base 12 (144 base 10), etc. In the middle they'd likely have 6, 60 base 12 (72 base 10), etc. As 12 neatly divides by 4, I wouldn't be surprised if they also had a 3 and a 9. Then you'd have 1, 3, 6, 9, 10, 60, 100, 600, 1000, as high as makes sense to go. I'd guess they'd be more likely to have 30's and 90's than 20's, but I'm just speculating wildly.