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That doesn't work because when the penny is hotter than the sun it radiates an equal amount of energy back to the sun.

The sun is a blackbody, it radiates light because it's hot. Once the penny is the same temperature it will radiate back at the sun until the two are in equilibrium.

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If the size of the penny was reduced to just a few atoms, wouldn't changing the number of atoms in the penny affect the rate at which the penny itself could emit radiation through blackbody emission? If so, I would expect there would be a size at which the penny could be small enough that it's temperature could grow arbitrarily large once the heat flux in = heat flux out from the penny.

I don't know very much thermodynamics, but I'm not sure that thermodynamic temperature is well-defined for a system consisting of "just a few atoms".

A penny at a temperature high enough to radiate the same amount of energy as the entire surface of the sun, would be at a far higher temperature than the sun.

If you surround the penny with sun with no gaps, all the light hits the penny.

If you add more sun, you add gaps, and the amount of light hitting the penny stays constant.

It never has more than one penny-surface-area of light hitting it, so it never has to get hotter than the sun.

Thanks, this is what allowed it to click for me.

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