the next most obvious base to use after binary
It seems to me you could make pretty good arguments for at least 3,4,6,8,12,16. Maybe 30 as well. You want a small base so you don't have to memorize lots of tables / build complicated circuits, but a large base so that numbers don't have too many digits; the tradeoffs here could push you to favour anything from 2 up to, I dunno, maybe about 20. You want a base with handy small factors because that makes some arithmetic tasks easier and introduces patterns into those tables that makes them easier to learn. Base 2 is particularly "natural" and has the advantage that single-digit multiplications can't overflow but base 3 scores well on that front too if you make your digits -1, 0, +1, and has the extra advantage that you don't need a special notation for negated numerals.
I think the small base argument dominates the large base argument for most use cases.
The main place the 'too many digits' argument carries weight, I think, is divisibility. It's handy to be able to express a third as a single number instead of a sequence that consumes every bit you can give it. With 60, you have short representations of halves, thirds, quarters, fifths, sixths, tenths, twelfths, fifteenths, twentieths, thirtieths, and sixtieths.
You pay for that in having a larger alphabet, of course, which the Babylonians cheated on by using tallies (real...
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