Really? Wikipedia says it is historically uncommon

EDIT: What language uses base-60??

Babylonians used it. That's the origin of our time counting system of hours, minutes and seconds and our angle measuring system of degrees, minutes and seconds. Anyway, if we someday change our number system away from 10, I hope the change will be towards a smaller base, like 8 or 6, because when one is a child, it sucks to learn the times tables. Base 12 would increase the number of entries by 44%. Base 8 would decrease them by 46%, and base 6 would have only 6*6=36 very easy to remember entries.

That's not a pure number system though, because it didn't have a separate character for each digit

In cuneiform they probably had.

Minutes in an hour, seconds in a minute, and something about degrees.