Pawpaw, Asimina triloba. I've had trees of this species for years now with mixed success. Basically I'd buy saplings and plant them in various places throughout the yard. Mostly the didn't survive the winter though and I'd have to start all over. I'm not certain on the reasons why they all died because I see images of them growing in full sun locations in yards, being used as a shade tree but it seems this really isn't where this tree wants to grow. It's likely the more sun they receive, the more water they require, and I just wasn't watering mine enough. When I finally got one to survive the winter it was situated beneath the shade of two oaks in the former vegetable garden before the oaks shaded that areas out. In nature this is an understory tree, sending up suckering stems to form its own grove. Cross pollination is essential to getting them to fruit, so once I'd found a successful location I planted a second sapling right next to it. I've read it can take up to 10 years before they flower but thankfully mine didn't take that long. It generally waited until it was 5' tall and will likely surpass that height this year. The second sapling is still only 3' and it's going to be a flip of the coin if it flowers next year which I hope it does because I'd love to finally taste this native fruit it's taken so long to grow. I believe you can get them to flower sooner if grown in containers.
The flowers are pollinated by carrion flies and beetles. The petals even resemble dead flesh a bit. Even with two plants flowering, I've read pollination can still be an issue. Those that grow them in containers swear by hand pollination which I might do to start. There was a tip posted somewhere saying to hang chicken skin among the branches... though that's just going to attract the neighborhood cats and other hoodlums to the yard.
Each afternoon as the sun goes down, little flies and things start showing up at the flowers. At a lecture with Eric Toensmeier, who is a reasonably big name in the Permaculture field, I asked him if it's worth it to plant things like Trillium viridescens and other fly pollinated plants beneath these trees. They actually flower at the same time of year. His response was really it's best to manually pollinate them yourself. And his reasoning makes sense, why risk a poor crop that you've waited the better half of the decade to try.
Another thing worth noting is that wildlife love the fruit even more that you do and often big bites are taken out of them before they're ripe. That's an issue I'll figure out how to tackle when that road comes. For now though I can simply enjoy the flowers... that smell of rotting fish.