With thanks to Florian Ingels, here is a paper discussing this problem.
We can define a coinage system $C$ as being composed of an ascending series of coin denominations $\{c_i\}$ with $c_1=1$. If the greedy algorithm always works to deliver minimum number of any coins for every target value, it is canonical (in the paper's definition).
There are a number of useful rules of thumb that apply to most practically-encountered coinage systems.
Clearly any two-denomination system $\{1,c\}$ is canonical. Using as many high-denomination coins as possible gives the smallest coin count.
For a three denomination system $\{c_1=1,c_2,c_3\}$, any problem with the greedy algorithm has to occur between the two-coin solutions involving the biggest coin, $c_3+1$ and $c_3+c_2$. So if $c_2=2$ the greedy algorithm will always work.
We can separate the problem for a coinage system $C$ into distinct challenges if for a given denomination $c_k$ we have $c_k\mid c_j$ for all $j>k$; then the system is canonical if the two systems $C]_{c_k}=\{c_1,...,c_k\}$ and ${}_{c_k}[C = \{c_k/c_k=1, c_{k+1}/c_k,...,c_n/c_k\}$ are both canonical. So for the US coin system $U=\{1,5,10,25\}$ we can split into a cent-based coinage $U]_5 =\{1,5\}$ and a nickel-based coinage ${}_5[U = \{1,2,5\}$ both of which are canonical.
For the counterexample of $A=\{1,5,10,12,25\}$, no split is possible. The paper cited earlier gives that a non-canonical system will have some value that can be expressed in two coins that the greedy algorithm will use more coins for. Here as you identify, $5+10=15$ cents will have a wrong greedy solution of $12+1+1+1$.
The paper "Optimal Bounds for the Change-Making Problem" (by Kozen and Zaks, TCS 1994) gives a bound of $x < c_M + c_{m-1}$ where $x$ is the counter example and $c_m$ and $c_{m-1}$ are the largest and second largest coin. They claim the bound is optimal. (I just browsed the paper, and I did not take the time to understand whether this automatically means you cannout expres it in the largest coin value)
Jeffrey Shallit (famous for his paper "What this country needs is an 18c piece") in 1998 posted a bibliography on the subject: mathforum. Shallit adds "... it is a problem in which it is notoriously
easy to make wrong conclusions -- some of them have even been published".
Good luck with your investigations.
Best Answer
I thought a google search would find an answer. To my surprise this turns out to be a pretty deep question.
From https://cs.uwaterloo.ca/~shallit/Papers/change2.pdf
The reference is from D. Kozen and S. Zaks. Optimal bounds for the change-making problem. Theoret. Comput. Sci. 123 (1994), 377–388.
From the same paper however: