Solution:

The amounts of money that you can have are all positive integers except $1, 2, 4$, and $7$. We can see that all these are impossible. To show that all other amounts are possible, it is clear that $3$ and $5$ are possible. Also, $6, 8, 9$, and $10$ are all possible, since
$$\begin{aligned} 6 & = 3 + 3, \\ 8 & = 3 + 5, \\ 9 & = 3 + 3 + 3, \\ 10 & = 5 + 5. \end{aligned}$$
Now, any higher amount of money has a remainder of $0, 1$, or $2$ when divided by $3$. If it has a remainder of $0$, it is $9$ plus a multiple of $3$; if it has a remainder of $1$, it is $10$ plus a multiple of $3$; and if it has a remainder of $2$, it is $8$ plus a multiple of $3$. No matter what, it is one of $8, 9$, or $10$ plus a multiple of $3$, so we can just add more $3$ coins to get from one of $8, 9$, or $10$ to our number. Thus, all higher amounts of money can be made.