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In other words when I open a music in a sound editor (eg Audacity) I'm presented with a waveform. Then I can raise the volume up to a point where the wave turns into a full block. Why can't the amplitude increase longer than -1 to 1 so I can keep raising the volume? AKA why is there a limit on the radio/TV volume?
The same sound after a very high amplification.
This is to do with the way that audio is stored internally to the digital audio workstation. Every additional "bit" of resolution is effectively multiplying the voltage measurement by 2, which is 6dB of gain. Just remember here, that when you are increasing the level of the signal - you increase the level of everything - including the noise floor. Now there isn't really much point in increasing the noise floor, so you really only need to concern yourself with working with a resolution that can handle an adequate "signal to noise ratio". The maximum resolution that a 64-bit DAW computer can operate with is 64-bits which is well beyond the resolution of the human ear. Really no need to go beyond this.
Digital audio represents sound as hundreds of thousands of amplitude samples over time. Each of those samples is stored in specific number of bits, usually 16 for commercial release or 24 or 32 for production. A 24-bit integer has a limited range: -2^23 to 2^23 - 1 (note that Audacity is normalizing these values to a range of -1.0 to 1.0). When you apply gain to a digital audio signal, you are multiplying all of the samples by some factor. If the result of at that calculation is greater than the maximum value for a sample, the value of that sample is clipped to the maximum value. As you try to apply more and more gain to your signal, more and more samples are clipped to the maximum value, giving the "block-like" waveform you posted in the question.
Volume limits on analog hardware knobs are different; they are an issue of the maximum power that a speaker amp is capable of outputting and the maximum power a speaker is capable of receiving (and the maximum volume levels before which sound becomes very, very unsafe).
There's an important difference between the amplitude of the digital signal (which has a maximum of 0dB because of how digital audio is stored), and the volume of the sound when played. It's also important to remember that in digital, the measure of dB is not a measure of loudness or volume, despite how it may seem.
Stored digital audio is essentially a template for how to reproduce a sound, and not the sound itself, so given the required power, you can play ANY digital audio (as long as it isn't empty) at ANY volume.
If you try to push a digital signal past 0dB you get that "full block" effect referred to, which distorts the audio. That can produce a desired saturation effect, but for the most part is a bad idea. It's better just to increase the power of the amplifier you are using (by turning up the volume knob on your TV or Radio, for instance).
You've skipped between a few things here. Music and TV sound and radio have different broadcast standards. Graphical wave forms are not a good way to read how loud something is, it only tells you the dynamic range of the audio. What do your meters look like? But very quickly, Digital audio cannot exceed 0dB which it looks like what your trying to do. TV audio has very strict limits it used to be compressed to -10dB; very poor dynamic range. This has now changed though to a different standard. What exactly is the problem your having?
Another way to look at this is that there's a limited number of 'steps' in the digital representation of a sound. For 8-bit audio there are 256 possible values that any sample can take on, for 16-bit there are 65536, etc. In no case is there an infinite number of steps, it's always finite.
When you raise (amplify) the sound level by one step, samples already at the peak have nowhere to go -- every sample that was one step below the peak, and every sample that was already at the peak, have the same value. Clipping has occurred.
Continue that process for N steps (where N is the number of possible steps) and you can see there's a finite amount that the sound can be increased -- all samples end up at the same (maximum) value. As a result you've decreased the dynamic range to zero. There's no more signal because there's no difference between samples.
Of course that's simplified. But the logical result remains the same -- each amplification reduces the number of unique values, it doesn't increase them. Infinity is not possible here.
