e^x is not single valued though. For simplicity lets consider general powers; a^b is only single valued when b is an integer. For example, if a^(1/3) = x, then x^3 = a. This equation has 3 solutions in complex numbers. Thus a^(1/3) is ambiguous as it describes the 3rd roots of a. Of course we can remove this ambiguity by defining a^(1/3) always denote one particular value of these roots but this does not get rid of the fact that there are many possible values which can all qualify as a^(1/3).
Substitute a=e and you'll see that e^(1/3) also suffers the same ambiguity.
The difference between DJMayes approach and mine is he is saying that e^x is defined as its power series. I don't think this is so. I think e^x is defined in the same way as a^x; By repeated multiplication; e^1 = e, e^2 = e*e etc...