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.

Use gray code.

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.

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.)

Because it makes it so that the number four has another meaning that, for whatever reason, people can't get past.

I've tried to explain it to people before, but we never manage to count past four without someone laughing.

Then they're missing a far better number: 132