Even since we started becoming conscious of having a use for the stuff lying around us, we found we need to keep count of them. The obvious tally machine was our fingers, and so the digital age began!
With only 10, it appears that the upper limit is rather limited, but maybe not! Conventional finger counting is linear, in that each finger represents just one unit. However, all numeric systems extend this by either using extra symbols for larger numbers, as in Roman numerals, or positioning the same symbols as used for units to represent powers of 10, as in the Hindi-Arabic system mainly used today. In computing, all calculations are in binary (powers of 2).
When writing down numbers, it is easy to visually see what the number is, up to very large numbers. With finger counting, all possible numbers must be shown with just 10 fingers, meaning that any positioning has to be within one's dexterity, and one's mental capacity to recognise the permutations.
A simple pseudo-linear extension of linear counting is to let each finger on one hand be a unit, while the other hand each finger is 5, allowing the first hand to start again. A simple extension of that is the flip a hand over, allowing each to count from 0 to 10, technically making 112 = 121 combinations, though most people would probably simplify it to just 100, by only counting to 9 on the units hand. That is still fairly simple to get one's head around.
A quantum leap is to let each finger represent a power of two, by folding down or extending a finger to represent 0 and 1 respectively. That gives 210 = 1024 combinations, 10 times more than the the previous example. However, the real quantum leap is to get one's head around using the strict power representation of each finger, which is a radical departure from most people's finger counting usage.
The scheme I am presenting here uses seven positions for each finger and thumb, giving a total of 710, or over 280 million combinations. Some may be dextrous enough to have more combinations, but these are what I could reasonable do, while still being able to operate each finger separately.
I must note that I have never actually used this scheme to count anything, as it doesn't provide enough benefit to try to train my thinking and hands to cope with the counting method. Also, even if I were to be able to handle counting at an average of a very optimistic one per second, it would take almost nine years of continuous counting to reach the end, let alone the severe repetitive strain I would have. This is just theory!
I leave it up to others to prove the effectiveness of the scheme. It was just something I happened to ponder upon, though I have probably spent more time writing this article than on thinking upon the scheme.