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Here is something I've been playing around for a couple of days:

Cart #paletteplay-0 | 2020-10-30 | Code ▽ | Embed ▽ | License: CC4-BY-NC-SA
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This cart allow to use a "higher" colour count palette for PICO-8, and of course, there is a trick here: You only have a canvas of 64x64 area as a group of 2x2 pixels are used for each pixel you can display using dithering.

This is mostly a proof of concept, I know I'm not the only one trying this. The code can probably be optimised, and the palette themselves are probably not perfect.

The bonus is that I have 3 variante of the palette:

The first one with 10 colours per channel eating the full first tile bank:
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A 8 colour per channel version:

Cart #paletteplay-2 | 2020-10-30 | Code ▽ | Embed ▽ | License: CC4-BY-NC-SA
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And a 6 colour per channel version:

Cart #paletteplay-1 | 2020-10-30 | Code ▽ | Embed ▽ | License: CC4-BY-NC-SA
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The difference between each is how much tiles are used for the palette.

I think the code is self explanatory, feel free to use any of these palettes if you see fit and to credit me if you use it :)

PS: cpc stand for "Colour per channel"

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Bonus:

16 colour per channel, use the wholme 128x128 tile area :D

Cart #paletteplay-3 | 2020-10-30 | Code ▽ | Embed ▽ | License: CC4-BY-NC-SA

To look better I should probably apply a gamma curve on the palette before conversion but too lazy. this is just a PoC
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More bonus, these palettes should work a bit better as there is some gamma applied to them:

10cpc:
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8cpc:
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6cpc:
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