How many days can a diabetic go without food AND insulin?

The liver produces certain amounts of glucose naturally. In many type 2 diabetics the liver acts like a sugar factory on overtime. Whether the individual was type 1 or type 2 this would be dangerous. After a certain amount of time spent fasting the body starts to burn its own fat stores to provide energy which would also cause blood sugar to rise and cause DKA. You probably wouldn't see a low because the basal is taken to deal with the natural production of glucose by the liver throughout the day. Blood sugar would rise and that's never good.