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Notably the physical limit is

https://en.wikipedia.org/wiki/Landauer%27s_principle

it doesn't necessarily take any energy at all to process information, but it does take roughly kT work of energy to erase a bit of information. It's related to

https://en.wikipedia.org/wiki/Maxwell%27s_demon

as, to complete cycles, the demon has to clear its memory.

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Does it not take energy to process information? Can any computable function be computed with arbitrarily low energy input/entropy increase?

No, and yes, so long as you don't delete information.

Think of a marble-based computer, whose inner workings are frictionless and massless. The marbles roll freely without losing energy unless they are forced to stop somehow, but computation is nonetheless performed.

I don't know how to compute with marbles without mass and stopping. Marble computers I've seen rely on gravity and friction, though I'd love to see one that didn't.

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