Add a layer of mini marshmallows. Leave enough space for the peppermint pieces at the top.
Crush the candy canes (add candy canes to a small freezer bag, seal the bag, and then pound the bag with a rolling pin). Be careful not to crush the candy canes so much that they turn into a powder. You just want chunks of peppermint in the top of the jar.
Add the peppermint chunks to the top of the jar and screw on the lid.