You can use a 16 bits or even 24 bits image map, but it has to be a color image. For example, you can use red for the lower 8 bits of the displacement, and green for the higher 8 bits.
Two component nodes can extract the red channel and the green channel - multiply the extracted green channel by 256 and add it to the red channel, and voila - 16 bits grayscale. Add blue, and you can go to 24 bit.

The pen is mightier than the sword. But if you literally want to have some impact, use a typewriter

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