Android Open Source Project, i.e. the base Android OS without manufacturer or Google additions.
As for why nobody is selling a pre-rooted AOSP phone cheaply, a few thoughts:
* phone manufacturer is a low-margin business for most. Insist on "cheap" and you need to sell a lot of devices to break even.
* AOSP means no Play Store, no Google services (so Maps etc won't work). You may see that as a positive, but the general phone-buying public won't so you just reduced your market dramatically. And Google don't let manufacturers pick and choose what they install, there's a bundle that comes all together.
* And you don't pay extra for the Google stuff being on the phone, because Google make their money from you using it. Hence not having it does not make a phone cheaper.
* But it's rooted, so you can just add gapps yourself? Sure, but the fraction of people who will put up with that hassle is tiny (and your shoestring company is going to get a lot of support calls from those who fail).
* So it's not for the general public, fine. It's also not for those who do online banking or phone payments (your market keeps shrinking). And if it's pre-rooted, so the service provider knows that any phone of that model is rooted, are controlling outfits like VZW going to allow it on their network?
The bottom line is that such a phone is a niche product which requires more than average support. That's either going to be expensive or follow the Chinese fake business model (older components, poor manufacture and no support). Or fail before getting onto the market, as some attempts have done.
If you want a phone without any of that stuff your best bet is still to find something that can be rooted and has decent ROM support and do it yourself.