As MattII commented, the Babylonians used Base 60 - which we still use! Anyone look at a clock lately?
(I first read about this in an Isaac Asimov essay; no, I don't know the title or where you can find it, I read the essay ~30 years ago.)
I'm not convinced that Base 60 is all that useful, but the advantage it has is that so many other Base counting systems are multiples of it (Base 2, 5, 10, 12, 20, any others you want like 3, 4, 6, 15, 30 as well).
I suspect that the Babylonians hit upon 60 (by say stretching a base-20 system for instance) for calendrical reasons--six sixties is nearly the number of days in a year. Also 60 is nearly the same number as 64 which is 8x8, so if they knew or were descended from a people who routinely used base 8, they'd be impressed with the all-round power of a base-60 system to come very close to unifying everything.
But where they came up with the 7-day week, which throws everything off, I have no frakkin' clue. Except that the lunar month is closer to 29 than 30 days so rounding it down to 28 and dividing that by 4 might have been right up those wacky Mesopotamian astrologer-priests' alley...
Octal has the advantage that it would be very easy to convert computer binary output into human readable octal numbers.
And I gather that sometime between say 1930 and 1990 someone or other tried to implement a human-comfortable form of built-up binary for casual counting and arithmetic. It features in Fred Pohl's The Coming of the Quantum Cats which was published in the late 1980s. In the set of alternate universes which proved to be technically dominant by the end of the story (and thus shut down the recent experiments in a set of somewhat less advanced ones in cross-time transport, since jumping sideways in time apparently had undesirable repercussions) the people from those timelines, or at least one of them, had such a system, using simple (rather silly-sounding!) words for binary counting up to base 8 or 16 or so, then presumably compounding bigger numbers using digits of these ("oodle-poo, toodly-oo, oppie-pop" were the way I remember them sounding). It just struck me as the sort of thing from OTL a guy like Fred Pohl would learn about and be struck as an interesting and possibly useful way of teaching people arithmetic. Perhaps a superior system of basic arithmetic put some of these allied advanced timelines ahead causing them to invent the cross-time switcher earlier; perhaps an early timeline that was advanced for quite other reasons happened to discover a neighboring timeline that used it and it seemed to the primary high-tech line like a good idea to adopt generally.
But yeah, from the replies to this thread it seems like base-10 is actually a rather parochial number base to have adopted!
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