The common system of simply holding up one finger for each item being counted is very limited, as one could only count up to ten items. The binary place value system can go up to 1023, but it is not not as practical, as it is more prone to error and requires tedious translation. My system is based on
the soroban, or Japanese abacus. It is easy to translate and goes up to 99. Each finger (excluding the thumbs) represents one (like "earth beads") and each thumb is five (like "heaven beads") the right hand records the ones place and the left hand the tens. For example, three fingers on the left hand and two fingers and a thumb on the right hand would represent 37.

[Spacecayote] Again, it requires some translation, so it's really only better if you expect to get over 100 or if you only have one hand free. Binary is still better for certain things.

There are many finger-counting and
finger-maths systems, many of them
more versatile than this. A common
system is to distinguish between the
three segments of each finger, and to
point to them with different fingers on
the other hand to get a multiplicative
effect.

Your invention sounds very much like
"Chisanbop notation", a finger counting
system based on the soroban. See
Wikipedia.

I have been using exactly this system for years after seeing it referenced somewhere.
And [Phundug], no hard surface is required, simply extending and retracting the fingers is perfectly suitable.

I tend to use fingers when I am tracking one set of numbers visually or audially and indexing things manually. Just an example, and not the best one, doing retail sales (without a computerized cash register) I do the addition for payment and change in my head, and might count the total number sold manually. And [max] if you can track more than one, or at most two number streams mentally, you have an extremely rare talent.

Another example is when you're figuring out the number of days between two dates: You mentally say to yourself "Monday, Tuesday, Wednesday..." but you meanwhile count 1,2,3,... on your fingers.

// if you can track more than one, or at
most two number streams mentally, you
have an extremely rare talent.//
Everybody can track two - one "visually"
(seeing the numbers in your head) the
other "audially" (hearing the numbers in
your head). Doing more than two is,
indeed, difficult - but then I don't think
you can keep more than two tallies on
your fingers either.

Finger maths is more worthwhile, but
finger counting is a bit like mouthing
the words as you read, shirley?

Long ago there was a local Chinese take-out and all the computations were done by hand (including figuring the "hot food tax" - it was California). I was wary of the accuracy and always checked against a calculator.

Day-uhm, they were always correct to the penny and could even make change correctly when given $17.20 for a meal which cost $16.95. The amazing part of all this? They could make correct transactions faster than the kids at McWendysKingBell using the latest registers.

[Spacecoyote] - My watch displays the time in binary and I'm now really quick at reading binary, though being a ZX Spectrum owner I was already a bit handy. Next idea: hexadecimal watch.

Now, if you could only control your toes.... then you could count to 9,999 on your digits.... and if you took viagra, you might be able to count to 19,999.