I just thought about how to find the optimal values for money units, given that you want your currency to come in $n$ different values (e.g. Euros come in 7 values for bills and 8 values for coins, so 15 values all together).
Fix a set $A\subseteq{\bf N}_{\ge 1}$ of size $n$ with $1\in A$ which represents all values your currency comes in (for simplicity I assume that there exist only coins and your currency has only values in the natural numbers, so no e.g. Eurocents). Let $f(A,k)$ denote the minimum number $j$ such there exist $a_1,\dots,a_j\in A$ with $\sum_i a_i=k$, i.e. we can do a payment of value $k$ using $f(A,k)$ coins. Further fix a discrete distribution $P:{\bf N}_{\ge 1}\rightarrow{\bf R}_{\ge 0}$ which denotes the probability that a random payment is of value $n$. Then let $g(A,P)=\sum_i P(i)f(A,i)$ be the expected value of the number of coins you need to do a random payment.
Are there reasonable values known for $n$ and $P$ (e.g. $P$ equals the geometric distribution) such that we can find an $A$ which minimizes $g(A,P)$, and if yes, is $A$ unique?