Taylor's exponential function series: an intuitive interpretation

Understanding how the exponential function works is like putting the last piece in the middle of a puzzle: everything in calculus and algebra makes sense. Not only do a lot of things in real and complex analysis come from the very definition of Euler’s number and its consequent magic properties, but also things everybody uses in his mathematical life, like taking a real number-power of something, have their ultimate definition in $e^x$.

For example, how do you calculate $2^\sqrt{2}$ or $3^\pi$ without knowing how the power operation with a non-rational exponent is defined?