Pawpaw

Asimina triloba
This native, small-to-medium tree grows well in the forest understory and forest edges.  The name pawpaw is a derivative of the papaya tree grown in the tropics. Its similar fruit is the largest edible fruit, about 3-6 inches long, native to the United States and has a custard texture that tastes similar to banana, mango and pineapple. The fruit has a very short shelf life and is not usually sold in stores.  The twigs and bark have a natural insecticide which make them somewhat resistant to diseases.  However, they do produce a lot of litter from the leaves and fruit. You can reduce this litter by planting just one tree to decrease cross pollination and thereby producing less fruit.
Wikipedia, University of Kentucky, Dept of Horticulture