What is the maximum number of SHA-1 hashes?

Clearly since SHA-1 hashing produces 40 characters each time, there is a finite number of possible hashes—does anyone know exactly how many?

SHA-1 hashes have 160 bits, so 2¹⁶⁰ of them.
(2¹⁶⁰ = 1461501637330902918203684832716283019655932542976 ~= 1.46 x 10⁴⁸)

Note that since you have a much larger message space than possible hashes, collisions are bound to occur.

Also note that the probability of collision is much higher than you might think. At just 2⁸⁰ messages the probability of a collision is 50%, thanks to the Birthday paradox. (ie: with just 23 people the probability that 2 people have the same birthday is 50%).