jmc

The height in millimeters of any stack with an odd number of coins has a 5 in the hundredth place. The height of any two coins has an odd digit in the tenth place and a zero in the hundredth place. Therefore any stack with zeros in both its tenth and hundredth places must consist of a number of coins that is a multiple of 4. The highest stack of 4 coins has a height of $4(1.95)=7.8\text{mm}$, and the shortest stack of 12 coins has a height of $12(1.35)=16.2\text{mm}$, so no number other than $\boxed{8}$ can work. Note that a stack of 8 quarters has a height of $8(1.75)=14\text{mm}$.