Easy, just use unary and draw a horizontal line that connects all the vertical lines.
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I can count to 4095 with only the index fingers and middle fingers of my both hands:
A bent phalanx means 1 and a straight phalanx means 0. I can bend those independently (including my most distal phalanges) on my index finger and middle finger left and right, so that is 12 bits 
Edit: I just tried and if I try really hard, I’m sort of able to include the phalanges of my middle fingers, pinkies and thumbs, and I can bend my wrists as well, theoretically reaching 30 bits = 1,073,741,823. It is very uncomfortable though.
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Abandon discretization (and discretion, for that matter).
I carefully position my outstretched index finger and precisely measure its angle with respect to the direction of the local gravitational gradient. Applying the tangent function on that angle allows me to express any number among the uncountably infinite continuum of the real number system.
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A go board is a convenient to store a number in base three. For instance the number 2022 (base ten) is 2202220 in base three, which is encoded by the following position.
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Maybe a capture could have an operation equivalence?
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For a project I’m working on I wanted a compact way to store small boards as base64 strings. So naturally I just converted from base3 to base64 
But then this turned out to have some drawbacks so I switched to doing it in blocks of three, converting each group of 3 intersections into a character (thus using only 27 characters unfortunately).
I’d like to improve the compression in the future (go positions are not exactly random so there are some bits to gain), but actually a few extra characters really doesn’t matter much in this case, so I’m sticking to the boring but practical solution for now.
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Here is the number of legal positions on a 19x19 board represented as a base-3 number:
(unfortunately not legal itself, would have been quite the coincidence if it was - the probability that a random position is legal is about 1 in 81)
More info on John Tromp’s page
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