Apparently they grow all over a good chunk of the U.S., but they’re too delicate for sale in a grocery store. I’ve never seen them grow wild or for sale anywhere. Any of you eat them? Where did you find them?
We bought a pair of pawpaw trees from Stark Brothers nursery. We planted them a year ago last spring, and depending on the variety, it might be 2 to 3 or 5 to 7 years until they bear fruit.
Or you could wander through woods, looking for a tree.
My old boss has a couple pawpaw trees and each year gives me a call to collect his harvest. My wife loves them and she’s one of the few people who does. I don’t think they’re bad but they are an acquired taste; sort of a custard consistency and sweetish-bitter taste. This past fall we had a ton and I made a pie from them which came out pretty well. My wife is from Peru and says they taste a lot like some native fruit down there and was excited as hell when she first saw and tasted them. I gave a spoonful to my three year old once and he made a face and started crying. So your experience will probably fall in the middle somewhere.
As you mentioned, they’re delicate for a grocery store and they unfortunately look uglier the more they ripen. When they look half decent (firm, greenish-yellow skin) they’re not ripe and when they’re ripe, the exterior looks rotten and blotched. Aside from owning your own trees – and you need a pair and I understand they can be finicky to bear fruit – you’re best bet would be to find them in the wild or make friends with a tree owner. Luckily, they do produce a lot of fruit though, more than any one person would need in a season so maybe they’d be generous 
Edit: When I saw my wife is one of “the few people”, I mean in my own circle. I’m sure they have a loyal following in the greater world. My old boss always gives us a call to collect them because he knows she likes them and because otherwise raccoons climb the trees and bust up the branches trying to get at all the fruit.
In the rocking chair next to Mawmaw’s, of course.
Might be the cherimoya (Annonaceae - Wikipedia), or some other custard apple (Custard apple - Wikipedia). Pawpaws are in the same family.
There are probably hundreds of topics where I think “I ought to ask Dopers about this,” but never think of it when I’m actually near a computer. Pawpaws have been on that list for a while, but I was prompted by a NYTimes recipe for pawpaw pudding Pawpaw Pudding Recipe.
Way down yonder in the pawpaw patch.
I’ve seen them at farmer’s markets.
Or just wander into the woods and pick your own.
But you don’t need to use the claw when you pick a pear of the big pawpaw.
The local “international market” (mainly Asian/South American plus US stuff) will carry them in season. They have an large and excellent produce section that puts US grocery stores to shame. See if you have something like that in you area (at the right time of year).
They grow wild all over the DC area. I have heard they are in Carderock, Scott’s Run and Rock Creek Park. I have seen them growing in Prince William Forest park. I would not think any fruit would be left on the trees this late in the year.
Win.
I’ve never seen paw paws for sale in a market. We’ve got two trees that have been bearing for several years (you typically need two different varieties in order for them to set fruit) and there are several weeks in fall when ripe fruit can be had. The trick is to get to them before the varmints* do.
I have to disagree with the poster who commented on taste. They do indeed have a custardy texture, but it’s a rich sweet flavor without any bitterness. I scoop out the fruit with a spoon while avoiding the large seeds.
I’ve never seen another fruit or vegetable that looked so ugly when ripe yet tasted so good. The trees can also have excellent yellow fall foliage, and weird brownish-purple flowers in spring.
*i.e. raccoons, woodchucks, Labrador retrievers.
I saw it said you were in the DC area, but near where I grew up is this: http://www.ohiopawpawfest.com/ - a yearly festival to celebrate the pawpaw. Cool stuff.
On the ends of leglegs.
In a cemetary in Monroe, GA.
Down south “Paw Paw” is a pretty common term for “grandfather”. Confused me no end as a child when I heard the lyric posted by Johnny L.A. Even as a youngster I knew that my Paw Paw didn’t come from no patch!
Probably fairly close to the prickly pears.
If you prick a raw paw, well, next time beware!
For info, this is Pawpaw Central: