My favorite thing to do with leftover potatoes is to make small patties with mashed potatoes and cheddar cheese, then either pan fry them until lightly browned, or crisp them up in a waffle iron and top them with a fried or poached egg and some green onions or chives. You can also press them into a muffin tin lined with prosciutto or bacon and top with an egg, salt, pepper, and chives, then bake to make a potato/egg cup. So many ideas! Here's a nice long list.