As I understand the problem, you are just wondering how many distinct signs humans can make with their fingers, not actually wanting to assign numbers to the signs. There's a ton of things you can do to get many more distinct signs than a few thousand. I mean, just learn sign language and spell out numbers. That will get you as high as we have words for numbers.
Other things that aren't disallowed in the rules:
You can add several different motions to your signs to make them distinct.
You can make more than just a few distinct, easy finger positions (not many people can independently change each of their 3 joints on every finger). Easy ones would be up, down, completely curled, finger pointing down, horizontal, straight 3/4 up, straight 3/4 down, curled but with the first joint straight (first knuckle pointed up), etc.
You can use the interaction between hands and body - i.e. pointing at a particular place on your body.
You can hold fingers together, apart, or slightly apart.
You can just make several different signs in a row. I hold up 1 finger, then 3 fingers, then 9 fingers, then clench my fist, then quickly loosen my fingers before clenching my fist again, and I've just counted to 13,900.