I did some study on this. Here is my two cents.
番, Fan1, is a measurement word for a piece of sheet, for example, paper, gold sheet, or land. 金薄十万番, 吏以纸万番赆之, 一番江水一番鱼.
番，when used in calculation，means the base number. There are 2 cases: 加番，翻番. For example, if the area of a piece of paper is 100 square centimeters, then 一番 = 100 (cm^2).
翻1番=100*(1+1)=200,翻2番= 翻1番 and 翻1番 again. (100*(1+1))(1+1)=200*2=400, etc.
加 means to put in another piece of paper.
翻 means to turn the paper to the other side, like turn over a page of a book. So it means double.
I checked a couple of online dictionaries, there are for sure some misleading.