I should have been a bit more clear - obviously they need to know who I am, and they need to figure out billing, but they don't need to keep any long term records or use those records for anything else.

If your browser directly negotiates with the site, there would be no third party aggregating data. If federated, you could choose the payment processor (perhaps bitcoin?), and only companies (or distributed protocols) you implicitly or explicitly trust receive the data.