My paternal grandfather enjoyed doing arithmetic using base-12. That's perhaps not surprising, he was an engineer, and he lived at a time when Britain used £sd. The British currency was pounds, which consisted of 20 shillings each containing 12 pence.
And the number 12 pops up all over the place: between noon and midnight there are 12 hours, 12 months in a year, 12 signs of the zodiac, a dozen is used as a common measure of eggs, there are 12 inches in a foot, ...
He referred to base-12 as duodecimal. At school I had to learn the times table up to 12 x 12. And the English language even has special words for 11 and 12.
Part of the reason that 12 is such as nice number is that it has a lot of factors: 2, 3, 4 and 6. Compare that to just 2 and 5 for 10 (as in base-10). With lots of factors numbers that are common expressed as multiples of 12 have easy to calculate 1/2s, 1/3s, 1/4s and 1/6s.
To use duodecimal you 'simply' add two symbols for 10 and 11: for example, you could use A and B and so you'd count like this: 1, 2, 3, 4, 5, 6, 7, 8, 9, A B, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1A, 1B, 20, ... There 10 is the number we usually call 12.
It's possible that my grandfather was influenced by the 1935 book New Numbers: How Acceptance of a Duodecimal Base Would Simplify Mathematics, part of that book appeared in the Atlantic Monthly under the title An Excursion into Numbers.
Although it's unlikely that duodecimal will replace decimal in everyday use, especially since metric is used in place of imperial weights and measures across the world, and since the British pound was decimalized in 1971, other non-decimal base systems are in common use.
Computers use base-2 (binary), and programmers often use base-16 (hexadecimal). Vestiges of another computer base, base-8 (octal), still remain: aircraft transponder codes are four digit octal codes.