What is wrong with negative feedback?

I am not talking about the kind you get as a flaky seller, but as used in amplifier design. It just seems to me that a lot of amp designs advertise "zero negative feedback" as a selling point.

As I understand, NFB is a loop taken from the amplifier output and fed back into the input to keep the amp stable. This sounds like it should be a good thing. So what are the negative trade-offs involved, if any?