Long ago, an American and I were counting something (I forget what, probably occurrences of some motif in a sequence or something like that), and when the American looked at my hand, he said "hey, that's a clever way to keep count!" Rather than using fingers individually, I was using the lines that separate the phalanges on the fingers, and the tips of the fingers, to count up to 16 on each hand, or 32 totally. I've always done that, and in India I'm not the only one -- I think it goes back to Vedic times (in particular, I think I was taught some such thing as part of some ritual or the other, when I was a child). So it didn't occur to me that anyone else would be surprised by it.
Can one go higher than 32? I didn't think much about it until I read this User Friendly strip. By representing 1 with a raised finger, and 0 with a lowered finger, and using both hands, one could in principle go up to 1023. There are two catches: one has to be familiar with binary numbers, and the fingers have to move very independently. The second is the bigger problem for me.
But the other day I realised that one can go up to 256 (or 255) very easily, by modifying my phalange technique. Count from zero to 15 (rather than 16) on each hand, but instead of adding the hands, use one hand as a 16's placeholder -- that is, use the hexadecimal system. One still needs to be comfortable with hex, but it is a useful skill for anyone who programs computers.
I haven't actually started counting that way yet, but next time I need to count a number that is likely to be much greater than 32 but less than 256, I'll give it a try.
In theory, with sufficient independence of finger movements, one could do 255 in one hand, as follows: use the phalanges as the lowest-position hex digit, and use the binary readout of the finger positions (raised/lowered, omitting the thumb) for the next position. With 16 inter-phalange lines/fingertips, and 16 possible combinations of raised/lowered fingers, one can do 255 in one hand. And combining the two hands, one could then count up to 65,535. But that is certainly too much for my level of digital dexterity or mental arithmetic. (The latter would have been so much easier if the world had standardised on base 16 to start with. Using base 10 was a huge mistake, but is now one of those suboptimal choices that are frozen and irreversible.)
UPDATE 24/11: Though I linked the Wikipedia article on hex above, I didn't read it. From this section, it seems I'm not the first to think of counting to 255. It doesn't attribute the originator but says counting on phalanges is common "in South Asia and elsewhere". It attributes the idea of counting to 1023 in binary to Arthur C. Clarke.