Lots of varying responses. Let me go a bit more in depth to try to get a clearer-cut answer.
Someone mentioned to me that 3 dB is barely noticeable, not true, because I have done direct swaps with amps and notice a 0.5 dB change. But it could be very well be that I felt the change more than I heard, it has been a while since I have made that comparison. I guess what I need to do is the A/B testing that audioholic mentioned.
I responded to the individual saying that 3 dB is more than just barely noticeable, and theoretically 3 dB is twice as loud. When I wrote that, I was implying that all things remain constant. Say for example all you do is swap amps so that the second one you are using has double the power output (let's just say efficiency, etc of the amp are exactly the same as the first, it is just the power output that is different). With all things remaining constant, such as frequency, voltage, etc, I was implying that 3 dB will be twice as loud. Now I never stated whether this will be to the ear on the mic. So I guess it would be best to describe the decibel increase required to hear double the loudness and for the mic to read double the loudness.
All this is in theory. Of course, real world testing would be different because there are too many variables. This thread probably won't get a clear cut answer, but when saying "3 dB is twice as loud" or "10 dB is twice as loud", the question I'd like to find out is under what conditions for each?