The moment generating function
As you suggest, the fact that $e^x$ is used increasing is a useful property here. In particular, that lets you in some cases to prove apply Markov's/Chebychev's inequality to the random variable $e^{tX}$ in order to get exponentially decaying bounds on the tails of $X$; see e.g. Chernoff bounds. In principle any other positive increasing function could be used in the same way, which substantially improve Markov/Chebychev boundsbut $e^x$ is a particularly useful choice because it is especially well suited to studying sums of independent random variables, as noted already in Yuval's answer.
