While admittedly the binary system gives you 1024 (ahem, mblase) numbers, and one hand gives you 32 (Ahem, indeed), one can magnify the number grossly by making the base 3 -- trinary. By having the finger position be in the set of Down, Halfway, and Up (rather than just Down and Up) you essentially increase your maximum to 59049 (and 243 on one hand). Taking this concept to a fuller potential, we decide that a person is able to identify 7 degrees of erection for their fingers (thereby creating a septenary system for counting. This would then allow for more than 282 million numbers (16807 one hand). Machine aided, oboy, look out.

Of course, one could use a machine much more effectively by mapping out a high resolution image map of the hand, and store each pixel as an integer, accessing these when a given place on the hand is indicated (hopefully by a very fine tool).

And to answer the question, I can count as high as ten, or even as high as two.

If I was asked to climb a short wall, I could count on my fingers to hold me, but if asked to scale more than 12 feet, with slippery footing, I would prefer to count on a rope held by a trained belayer.