[anigbrowl]

No, you can represent it with your fingers but not count it. By 'count' I mean an ordinal traversal of the sort you teach kids.

[eru]

Use gray code.

[anigbrowl]

Good luck teaching that to a 4 year old, or someone who's barely numerate. Higher bases are better for learning counting because the person can count a variety of small quantities without needing to understand the concept of a number base. It's not that I don't approve of binary counting by hand, but it's hardly useful for beginners or primitive societies.

[eru]

Oh, for that application, my suggestion of gray code is even worse then binary.

Though actually, you are not really arguing for a large base, but for unary. (Or a mixed unary-large base system.)