That's the good part, you don't.
That's why you don't hear of people quantifying how much extra peak power you get from adding xxx farads of capacitance. Like most audio tweaks it has a basis in science but falls apart when you ask how the maths connect.
The benefit from doubling capacitance is halving ripple voltage. Although ripple is related to load current, the context, scale and goal here are vastly different. Definition of ripple is clear, definition of "peak power" is not. Also, you can measure the effect of ripple on the output, so once it is below the noise floor (like every good product measured on this website nowadays) then reducing ripple further has no real benefit. Maybe Amir can include output ripple noise vs output power measurement, who knows.
Consider the scale of things: Up to a certain duration, we can argue that "crest factor"... "program material" blah blah blah... they allow the usage of a power supply that does not have enough sustained output capacity for a flat square wave DC signal to still be able to reproduce the typical music waveform. But I ask to think in another way: You still need a known certain amount of power in order to reproduce the music signal, it's just that the shape of the input power can be different. And capacitors help to bridge that gap - The incoming power waveform can be totally straight DC (or it can follow the AC mains voltage, depending on context), yet the output power waveform is the music signal. The difference is being covered by the capacitor.
So two points to note:
1) For one output power waveform, regardless of how the input power waveform looks like due to capacitors and whatnot, the amount of power delivered by the power supply is still the same. The capacitor changing the waveform shape can allow for certain cost savings possibilities e.g. lower maximum transistor current, but the amount of power is still the same on average. Ergo, a 25W power supply is not going to produce a music that averages >25W no matter how much capacitance you give it. It can produce 50W peak... 75W peak... how long you want the peak to be, which leads us to the next point:
2) How long should the peak duration be? We have the
CEA-2006/490A standard: 1kHz sine wave signal that utilizes a 8 ms burst at full power with a THD+N <1% followed by 24 ms pause. This is repeated every 32 ms with a 6dB Power Crest factor. but this is not what I'm talking about here because notice the duration numbers: The period is only 32ms long. As long as you have enough capacitance to cover the duration, the bottleneck becomes how much power the power supply can provide every 32ms. How much capacitance is needed? Up to debate depending on crest factor and allowable distortion, but definitely
not in the order of a few seconds that is provided by extreme capacitor values.
Having extra power for this kind of long duration presents a different kind of issue if that power is being used for "better performance" or "moar power". I'm totally fine if this capacitance is there to prevent glitches, mains AC issues, and even to provide graceful shutdown - Things affecting actual functionality. But if you have "more temporary power for 10 minutes", then here's a problem in actual use: If you're playing song A which needs the extra power, and you decided to play song A again, would the second time you play song A run into power issues? Or how about 1 second: The first two bass drum kicks are fine, then the rest start to clip. Then after a while of recharge the next two kicks that come are good again, followed by clipping ones. This peak power is unusable if you ask me.
Moral of story: Provide just enough "peak" capacitance to sustain maybe one 5Hz or 10Hz cycle. (0.2s ~ 0.1s) Maybe 1Hz / 1s if you really want. Let incoming power supply deal with durations longer than that. Because any "peak" duration longer than that is going to cause performance fluctuation with a time frame that is perceptible by humans.
