Why is the number $e$ so important in mathematics?

The number $e$ is not very important in mathematics!

On the contrary it's the exponential function $$\exp(x)=\sum_{n=0}^\infty\frac{x^n}{n!}$$ that's important. The number $e$ is just $\exp(1)$ - just one value of the exponential function. Compare how often one sees the exponential function as opposed to $e$ by itself. In the four previous answers to this question, three focus on the exponential function, and only one on $e$ itself.

One could easily write entire textbooks about this subject, and I think the only way to properly appreciate the answer is just to learn a lot of mathematics. But I will mention one perspective which suggests that $e$ ought to be fundamental to mathematics.

This is the perspective of groupoid cardinality. The definition of a groupoid is slightly technical, but for the purposes of this discussion a groupoid is a collection of groups. In many counting problems involving groups it turns out to be natural not to count all of the objects involved, but to divide each object by the number of symmetries it has. For example, if you draw five dots in a line on a paper and then fold the paper in half through the middle dot, two pairs of the dots have been identified and one dot has one half identified with the other half; in other words, the folding symmetry has cut the middle dot in half, so one might say that we now have $\frac{5}{2} = 2 + \frac{1}{2}$ dots.

This is the basic idea behind groupoid cardinality: the cardinality of a groupoid is the sum $\sum_x \frac{1}{|\text{Aut}(x)|}$, where the sum runs over all the isomorphism classes of objects of the groupoid (here, over all the groups in the collection, where $|\text{Aut}(x)|$ is the size of the corresponding group). All groups here are finite.

Now here is the fundamental statement.

The groupoid cardinality of the groupoid of finite sets (e.g. the collection of symmetric groups $S_1, S_2, ...$) is $e = \sum_{n \ge 0} \frac{1}{n!}$.

In other words, $e$ somehow embodies a fundamental property of finite sets. It is possible to think about the whole theory of exponential generating functions in this way, in particular to think about $e^x$ and its derivative property this way. Part of this story is described, I believe, in John Baez's the Tale of Groupoidification.

One of the more concrete ways to think about the relationship between the symmetric groups and $e$ is via this generating function. The identity described in that blog post, which sometimes goes by the name of the "exponential formula," explains, among many many other things, why the probability that a permutation of a large set has no fixed points is about $\frac{1}{e}$.

One of the first exposures students have to $e$ is in the formula for simple interest compounded continuously: $$A = Pe^{rt}$$

This formula can be easily derived from the formula for simple interest compounded discretely: $$A = P(1 + \frac{r}{n})^{nt}$$

The derivation depends on the identity $$e = \lim_{x \to \infty} (1 + \frac{1}{x})^x$$

So although this may not be the most important application, it is one way in which $e$ is used and calculated.