I came up with a pi based counting system. The place holders are thus

... pi^3, pi^2, pi^1, pi^0, pi^-1, pi^-2 ...

The digits are 0, 1, 2, 3. So you count 1, 2, 3, 10, 11, 12, 13, 20, 21, etc just like you would count in a four based system. Note, however, that 10 etc are irrational, because they contain multiples of pi. It's a lot like a four based but pi instead. Since pi is irrational, the only "rational" numbers in this system are integer numbers of pi, and 1, 2, 3 themselves. But all other numbers come out irrational. Thus 4 (base decimal) is

4 => 10.2201..... and it goes on forever

I don't think it's possible to really count in this system. For example, you can't easily count the fingers on your hand because beyond 3, the numbers become irrational. So to you the number of fingers on your hand

=> in pi base 1, 2, 3, 10.2201..., 11.2201..., 12.2201..., 13.2201..., 21.201...

=> in decimal 1, 2, 3, 4, 5, 6, 7, 8

I had to calculate the conversion each time. In any case, I think it's interesting.