In the UK, we can have up to £1.43 in regular coinage without being able to "make change" for a pound: 50, 20, 20, 20, 20, 5, 2, 2, 2, 2.
Of course, this ignores the fact that we have the two pound coin which allows one to have an infinite amount of money without being able to change a pound. Thanks to heyoka for pointing this out.
There must be a neat algorithm or process for determining the largest amount of sub-unit given a set of available denominations, but it's too near the weekend for me to think about.