Non-answer: For $100/day, I wouldn't subject myself to such a dangerous challenge. The body is really dependent on having the right set of nutrients to function, and it becomes very unhappy when you take them away.
Answer: Soy or Quinoa. A particularly difficult class of nutrients you need are amino acids. There are nine amino acids you must eat because your body cannot produce them: leucine, isoleucine, lysine, tryptophan, histidine, phenylalanine, valine, methionine and threonine. Almost all vegetables are incomplete proteins, meaning they are missing one or more. Starchy vegetables, such as the potato, lack lysine. Remember lysine? It was the thing the builders of Jurassic Park kept as a last line of defense to keep the dinosaurs under control. Yeah, that lysine.
Not all vegetables are incomplete proteins. Soy and Quinoa are complete proteins. Now mind you I did not look at their other nutrient loads, but if you don't have your amino acids, bad things happened, so looking for complete proteins was a good start.
Also worth checking: can each food be eaten raw? Many vegetables are not edible until they are prepared. Potatoes and many grains cannot be eaten raw, because the body cannot process them until the cooking process denatures undesirable compounds for us. Potatoes can actually be toxic if eaten raw, and you don't get the carbohydrate energy from them because the human body has a hard time with unprocessed carbohydrates.