1mW continuous power can charge a battery albeit over a long time.

Err.. for simplicity sake, I'd assume the 1mW is net received power after transmission and conversion losses. And 1mW is also an arbitrary number for the sake of argument.
My argument that 1mW is sufficient to charge a battery is generally based on the design of solar powered night light.
Assuming the night light LED uses 3mA at 1.2V and needs to run for 8hours per night. That means simple total power drain is just below 29mW. Hence 3mW of continuous charging power for 12 hours per day during daytime will be sufficient for it's night time operation. It's a simple calculations but suggest plausibility. You can see why I think 1mW can charge a battery, just depends on what is the load and application.

Next consider this. Let's say you are transmitting data using some sort of RF signal from one transmitter to another receiver. Now what's to stop me from say making another receiver that also reads the same RF signal, but only converts the electrical signal I received to powers say an LED?
Both electrical signal and electrical power are made of voltage and current. Whether it's a signal or power is our interpretation and application. Of course in order to create the signal, we modulate data into the signal waveform (commonly called carrier). And usually signal lines doesn't carries much power due to well, it's not designed to transfer power after all.

No I mentioned 3mW can be use to charge a battery for very low power applications. I did not say it must comes from wifi. The question for me was merely, can 3mW or 1mW charge a battery?
I interpret 3mW or 1mW as available net power. You interpret is as before conversion losses. So it's a simple misunderstanding. That in no way means I do not understand regulator efficiency and battery self discharging.

And it's only a small pool of light yeah but I'm only pointing out a product available on the market, not arguing about the pros and cons of it. Also the example is meant to point out such low amount of power is only suitable for very low power application whereby stressing the key issue of wireless power transfer --> transmission losses, which in my mind has always been the bigger picture.

I'm a firmware programmer, which in itself is a field that covers programming and a little on electronics. I'm not strong in analog electronics though so I'm curious about the amplifier you mentioned. Since the purpose here is merely to collect power and not the signal data, is there really a need for amplifier and mixer? I'm assuming the right antenna will convert the carrier waveform into similar electrical waveform. Since it's an oscillating waveform, then theoretically we can use the voltage doubler or boost circuitry concept to boost up to the right amount of voltage and current we want right?
And if amplifier requires power to function and is required in order to extract energy from RF signals, it would be a chicken and egg story and doesn't explains why passive RF tags work.

Then theoretically we can use the voltage doubler or boost circuitry concept to boost up to the right amount of voltage and current we want right?