Does amplitude of FM signal even matter?

Thinking about FM modulated signals that store information only in frequency so that their amplitude has no effect on the sound, I began to wonder: Does the amplitude even matter in the end?

Well, mathematician would say - "Yes, it does. At 0 amplitude there will be no signal."

But I'm rather asking whether it matters, to the close receiver, if I use 10V or 10kV.

What would happen if I had used the former? Would it turn into Tesla tower? Does higher amplitude have better spread? If yes, how far can one go? (and how close to it then - literally)

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"...if I use 10V or 10kV as the...wavelength" - voltage is not the same as wavelength. – JYelton Jun 20 '14 at 7:40
It is not indeed. I must have dropped some word from that sentence since it makes no sense... – Tomáš Zato Jun 20 '14 at 8:19

Well it does matter a lot actually.

The only difference between a mathematician and an engineer is that the former would say:
"Yes, at 0 amplitude there will be no signal"
while the latter would say (possibly opening a beer can):
"Yes, if amplitude is below a certaing level your transmitter is useless"

The question you are asking is actually quite vast, you can get a degree in telecommunications, at least in Italy, but I'll try to stress the most important points.

Power
Power is proportional to amplitude squared. The more power, the more range, the more range, the happier you are. When you communicate via ideal apparatuses amplitude determines only the range: your receiver has a minimum power requirement, you know how much power you are sending, and you know that power decays with the distance squared. If you are close enough power does not matter... Theoretically. We don't use big ass transmitters as you suggest becaues when you exceed a certain amount of power your transmitter becomes quite difficult to be built, mantained and operated. And I bet efficiency drops too. So that's why we use many little transmitters spread on the hill tops. For the records, I believe 10kV is quite a common voltage range for transmission stations.

What we need to do now is ask: why is there such a minimum power needed to receive the signal? The secret lies in the SNR, namely the "Signal to Noise Ratio". Demodulating an FM signal is not a trivial task, but let's assume that you are doing so measuring the frequency deviation per each period received, i.e. you have a stopwatch and measure how long does it take to your signal to go from one maximum to the next one. You write down all the values you get, and you've rebuilt your original signal (some math needed but you pretty much have it). Now what happens if you look close? I hope you did see a signal on a scope at least once: you hook up a signal generator directly to your scope and see a nice sine wave waving at you, but if you look closely you see the bad guy: noise. White gaussian zero-average thermal-or-whatever noise. The problem is that electrons are ruled by statistic functions, so they don't always behave as you think, and you end up with a tremulous line instead of a nice one:

That kind of shaking is superimposed (aka summed) on your signal.
Luckily enough your sine wave amplitude is way higher than the noise peak to peak average amplitude (note the average word). When you measure the distance between two maximums you don't even see the fact that the maximum is not a maximum but is full of up and downs. But what if you go down in amplitude? What if you set your transmitter to use let's say $v_{tx}=2v_{n-pp}$ where $v_{n-pp}$ is the noise peak to peak average amplitude (you guessed it didn't you)? Well now measuring the distance between two peaks is difficult if not impossible, so here is where the minimum power requirement come. That actually is a requirement on the SNR: what matter is not signal power per se, but signal power over noise power.

So that was for your first question. Please note that power depends on the transmitter power but also of what happens in the middle: a nice antenna means more power in the air, less power dissipated on it. A nice path, i.e. free air, means less power dissipated bouncing between skyscrapers.

About the close receiver: when your SNR is over a certain amount improving it does not really make a difference.

I don't think you can turn a radio transmitter in a tesla tower unless you are a very bad designer.