Kitchen question

We live in an old house--over 100 years old. As is usually the case with houses of this age, the configuration of storage space leaves a lot to be desired, especially in the kitchen. I came home after buying groceries this afternoon and cleaned out the refrigerator to make room for fresh leftovers. When I was done, I had an empty vegetable crisper in the 'fridge and a pile of fruits and vegetables on the counter. I know that many fruits and vegetables do not need to be refrigerated, but in the situation in which I find myself--more refrigerator space than pantry space--which fruits and vegetables should I absolutely not refrigerate? Bananas. Any others?