This is probably the cheapest recipe I know.
What you need:
1/4 cup flour, maybe less
any seasonings you want, I use garlic powder, salt, and pepper
Shred the potato with a cheese grater. Squeeze out some of the liquid over the sink. Mix in some flour until it’s just a sticky mess. Heat up a skillet on medium and put a few splashes of olive oil on the pan. Make some patties with the potato, and fry just like a regular burger.
Serve with ketchup/catsup and toast (using any bread nearby).
EDIT: When you squeeze out the juice, dryer is better. It gets more crispy and isn’t as starchy.