Why do we have ten dollar bills if we also have fives and twenties? A single ten dollar bill could easily be replaced with just two five dollar bills and multiple tens could be replaced with a combination of fives and twenties.

Carrying a balanced combination of fives and twenties would work just
as well, if not better than, several tens. For example, rather than ten tens, have three twenties and eight fives, or four twenties and four fives. Having fives, tens, and twenties is just overly complicated.

Eliminate everything under the quarter before you start on the bills man. The PITA in making and carrying change is portaging the metal around. I usually carry zero cash relying on debit. Never have a problem if you think ahead.

The trouble is the ratios between denominations. You need lots of quarters because there are four of them to the dollar, and you need lots of singles because there are five of them in a five dollar bill. There are only two fives in a ten, however, so carrying two fives isn't that big of a deal. There are only two tens in a twenty, too, so tens are pretty useless. If you eliminate the ten, you're left with four fives to a twenty, which is a great ratio in my opinion.

Why don't we just eliminate all coins, all bills, and move to a complete debit card only system? Then you wouldn't have to carry any bills. And if we embed a chip in the back of your hand, you won't even need the bother of carrying that big ol' heavy card around.

Pretty much every currency I know uses 1,2 & 5 or those numbers multiplied by 10 or even 100. There must be a very good reason, probably involving the fewest number of items used the fewest number of times, to give all possible sums.

//Pretty much every currency I know uses 1,2 & 5 or those numbers multiplied by 10 or even 100. There must be a very good reason, probably involving the fewest number of items used the fewest number of times, to give all possible sums.//

I think this is largely because people thought more of the bills' individual uses than of the system as a whole.

I got it! We'll use binary denominations! We already
have the one and the two. I imagine banks will figure
out some way to screw people who can't read
hexadecimal.

You need a system that minimises the number of notes/coins that need to be used to make up any given amount. So, say you have a range of amounts that need paying, from 1 to 1000, and you count the minimum number of notes/coins you need to pay each amount - e.g. 1 needs 1x1 = 1 coin, 2 needs 1x2 = 2 coins, 3 needs 1x2 + 1x1 = 3 coins, 4 needs 2x2 = 2 coins, 5 needs 1x5 = 1 coint... and so on. Then you add up all the coins, and divide by 1000 - and then multiply by the number of unique denominations there are - giving an index number that describes the "efficiency" of the token system. The lower the better. It might be interesting to see how the general 1, 2, 5, 20 (25), 50, 100, 500 decimal systems of today measure up against older Œ, œ, 1, 2, 3, 6, 12, 24, 30, 60, 240 £sd system of yore.

The problem with that system, [zen], is that it
assumes that (a) you have a complete set of
coins/bills to hand and (b) you are equally likely to
need to pay any amount.

In reality, such a system is not optimal, except
under those very narrow conditions.

I suppose then you need a measure of how many ways you can use the available denominations to make up any given amount.
e.g.
If you only have denominations of 1p, there is only one way to make up 19p.
If you have denominations of 1p and 2p, there are 10 ways of making 19p.
If you have denominations of 1p, 2p and 5p, there are <some more> ways of making 19p.
Presumably, there's some way of normalising this property to a value between 1 and 0 that you could then merge in with the previously described measure.
That's based on the assumption that since we don't know how many coins of a particular denomination we migh have in our pockets at any one time, the higher the number of different ways of making a given figure should provide a greater chance of being able to make that figure with the coins available.

Trouble is, once you start allowing for being paid change, it matters less that you're able to make the exact amount, and more that you're able to pay and recieve change from a suitably stocked float using a minimal number of coins, which takes us back to the original measure of optimality being linked to minimising the number of coins required to meet a wide range of values. (I suppose you might add into the measuring process the idea that for example 19p while normally being counted as 1x10p + 1x5p + 2x2p = 4 coins - it could also be met by 1x20p + (-1)x1p = 2 coins - one of which is change)

So I do think that covers your limiting factor a) , while I have to concede that b), the fact that lots of things cost 99p is a factor probably covered by any of this at all. Having said that, is not the 99p thing a result/symptom of the token system, rather than one of the defining factors? In the days of shillings and pence, things generally did not cost 99p (I wonder if they had some similar effect though, say 11œd) so I'd argue that's an emergent result rather than something else, no?