The cheapest diet I ever tried was the first phase of the Fat Smash Diet. It was very filling because you ate a lot of oatmeal and brown rice, but you also were allowed unlimited amounts of all veggies except potatoes and avocados, as well as unlimited fruit. You could also eat cooked beans and egg whites. You cooked with very little oil. You could eat a small amount of dairy - up to 2 cups of skim milk or 2 servings of yogurt. However, you could not eat cheese. Phase 2 added small amounts of other foods, but you could stay on phase 1 as long as you wanted. If you ate the full amount of oats, rice, dairy, and a reasonable amount of veggies and fruits, the diet came to about 1300 to 1500 calories per day.
You were allowed 1 cup cooked oatmeal and 2 cups of cooked brown rice or other grain, so you get full on very cheap food, then surround it with whatever veggies and fruits you can afford to add to it.
Our Fat Smash forum is closed, but I think you can still access it directly at http://www.3fatchicks.com/forum/fat-smash-diet-211/
If you are interested, you can browse the threads for sample menus, etc. I would suggest starting from the last page as there are more menus and ideas there.