Python 2, 32 bytes
Try it online! Sorts the list in place.
The idea is to use a sorting key function without a
lambda. A good candidate was
x.strip, which takes the string
x and removes its the left and right edges all characters in the input. For example,
"abcdef".strip("faces") == "bcd".
quarter -> np dime -> npr nickel -> pr penny -> r
which are lexicographically sorted. I found the string
npr by brute force.
npt also work, and there are none shorter.