My record is 3124. I was in St Malo as a kid when binary fell into place for me (it was explained in my Usborne book on computers), and around then I started counting in binary on my fingers. Like this person (who also has a demonstration video) I still do it, and tap-count to pass the time.
A game I play sometimes is to see how high I can count on my fingers. Regular counting gives you 10. I used to just do 5 counting on each hand and get up to 25. My mum has always counted to 12 on one hand (using her thumb to count three segments on four fingers). That's pretty good because it leaves a hand free, and in theory you can get to 143 like this. The Babylonians counted using finger segments, by the way, counting to 12 (with one hand hand), 5 times (with the other), up to 60. Binary gives you 1023 (ten fingers in base 2, so the maximum is 2 to the power of 10 minus 1).
Switching bases is the right thing to do. If you count finger joints, you can get 5 states per finger (3 joints, plus the fingertip, plus the zero or not touching state. For the thumb you use 2 joints and the muscle at the base), which means you can count in base 5.
It has the disadvantage of taking both hands, but you can do it without a surface. Using your knuckles means it's hard to lose your place too. The "rest" position is hard to maintain (because it requires not touching fingers), but the binary method has that problem too. It's easier for higher figures.
Like this, I can get up to 5 to the power of 5 minus 1, which is 3124. It doesn't seem possible to keep counting in higher bases (and still be robust to slightly misplacing fingers) so I think I need to find a different strategy to count higher. I'd like to keep it to my hands if possible. Maybe counting in two differently based cycles simultaneously, and using their interaction to reach a higher total? For example, I could use my left hand as a counting board and count in base 20 along it with my right thumb, and in base 19 along it with my right little finger. Then, if they were coupled right, the difference between the two when one was at a fixed point could represent the position in the overall cycle? I don't know, that needs some thinking about.
Update: There's more.