Asimina triloba. Pawpaws are a fascinating species native to the eastern United States. They are in the "Custard Apple" family (Annonaceae), which generally grow in the tropics. However the pawpaw is cold hardy. The look of the big leaves and size/taste of the fruit is certainly tropical, but you can grow it here in Vermont!
They can reach approximately 8 feet high, and thrive even where it is quite moist. You will need to plant two different varieties to get a nice crop. In spring, the large but cascading branches are covered with creamy blossoms. The fruit is very good for you.