What’s the relationship between gain (dB) and power (watts)?

@rols

In my mind big power amps sound powerful because they are capable of making the speaker follow the music. They sound more exciting, not louder.

I thought that increased damping factor was part of the reason that higher-power amps have more speaker control (grip), but I'm probably missing other factors; interesting topic.

This is where my technical ignorance seems significant.  Maybe I’d need a textbook to understand this:

Of course, gain only works so long as you don't exceed the output limits, whose absolute limit in a linear amp is by the power supply rails.

This is nothing more than an unnecessarily techno-jargon laced way of saying exactly what I said: Gain is a multiplier, power is a ceiling.   

This discussion is getting out of hand as simple DC models are being compared with exotic AC ones. The math is being dumped out of a need for a model of understanding which doesn't match the challenge of the questions. The answer to all of them is... the questions are too difficult to separate into little pieces. The basics are just that, basics. The answers are beyond the understanding of most people because they encounter ideas such as statistics and psychology, which some people think do not "fit" into any discussion of music. However, they do... as well as other areas of mathematics and exotic theory, both electronic and as apparently "simple" as component design (so sorry, that is ALSO extremely math oriented). The bottom line is, most people simply cannot, will not, and don't either believe or know the answers. That means if they are right or wrong. Good luck.

@millercarbon

Gain is a multiplier, power is a ceiling.

a ceiling of what, though?  Current?  Voltage?  

Power.  

Seriously. Helps to think things through. What is power? Current? Voltage?  
Here I'll give you a hint: Watt is power? That's not a typo.