1965 US Quarter Dollar Coin With Extra Leaf!?

That is Machine Doubling. Machine Doubling affects the outside edges of the devices. It is not a doubled die. A doubled die shows a spread on the devices in the central area of the device, not on the outside edge. Machine Doubling is caused by the machine damaging the freshly struck devices. Machine Doubling can also happen on a doubled die as well. But they are still doubled dies, with Machine Doubling. Here is an example:

Note all the devices are showing a spread on the center areas of the devices? But note the yellow arrows. This is showing the areas that are affected by Machine Doubling. MD is usually in a direction. Note how all of the devices are affected with the MD in the same direction. That year a lot of the doubled dies had Machine Doubling. But they are still doubled dies. (with Machine Doubling)