I'd say your cheapest (and still flavorful) beer will be a Berliner Weiße.
You only need 1.038 OG (55% 2-row, 40% wheat malt, 5% table sugar - Plus ~1/2 lb uncrushed grain for souring your mash).
Mash (2-row and wheat) at 156F for 60 minutes, let it cool down to ~100F, add the grain (that contains the "beer-spoiling" Lactobacillus bugs) and keep at "Body temperature" for ~2 days. A beer cooler is great for this.
It will stink (keep it in the garage or outside!) but tastes much better than it smells.
Sample the flavor regularly until it will reach the sourness of your choice. After 2 days it will be sour enough. Now it's time for a mini boil (15 mins to kill the Lacto and give a tiny bit of bitterness to the beer). Add the sugar (~8oz for 5 gallon batch) 2/3 Oz 5% AA hops or whatever will give you about 5-6 IBUs of mini-bitterness.
Pitch an S-04 or S-05 or similar yeast at ~70F and ferment as normal light ale.
Can you get cheaper than that? Probably not!