Mildly Interesting

Just buy bananas that are of the ripeness you like and put them in the fridge. They'll easily last a week or two and tastes better too (my opinion). Don't worry if the shell turns brown, the insides are still good. Place them in some kind of container (plastic bag will do) when you store them in fridge and they'll probably last a month if not more (I wouldn't know, I always eat them sooner).

When storing in fridge, the low temperature slows down ripening but oxygen causes the shell to turn brown. When the shell is brown enough it will start to ruin the insides. The plastic bag, or other airtight or near airtight container, reduces the amount of oxygen that gets to the bananas. When there's less oxygen to turn the shell brown the bananas will last who knows how long. In room temperature the shell and the insides will ripen about at the same rate, and usually pretty fast.