The Euler's constant $e$ appears whenever an extremely small change (increment or decrement) is applied an extremely large number of times e.g. an increment of 1% (1.01) applied a 100 times. It is easy to understand this concept of applying a small change a large number of times using compound interest.
The formula for compount interest is given by,
where $P$ is the principal, $r$ is the yearly rate of interest in percentage, $n$ is the number of compounding periods and $A$ is the total amount at the end of 1 year. Let, $P=1$ and $r=100$.
If the interest is compounded annually, i.e. $n=1$, then
$$ A = (1+1)^1 = 2 $$
If the interest is compounded semiannually, i.e. $n=2$, then
$$ A = \left(1+\frac{1}{2}\right)^2 = 2.25 $$
If the interest is compounded quarterly, i.e. $n=4$, then
$$ A = \left(1+\frac{1}{4}\right)^4 \approx 2.441 $$
If the interest is compounded monthly, i.e. $n=12$, then
$$ A = \left(1+\frac{1}{12}\right)^{12} \approx 2.613 $$
If the interest is compounded daily, i.e. $n=365$, then
$$ A = \left(1+\frac{1}{365}\right)^{365} \approx 2.714 $$
If the interest is compounded hourly, i.e. $n=365\times24$, then
$$ A = \left(1+\frac{1}{365\times24}\right)^{365\times24} \approx 2.71812 $$
If the interest is compounded every second, i.e. $n=365\times24\times3600$, then
$$ A = \left(1+\frac{1}{365\times24\times3600}\right)^{365\times24\times3600} \approx 2.71828 $$
In the above situations, we observe that an extremely small change, say $(1+\frac{1}{n})$ is being applied an extremely large number of times i.e. $(1 + \frac{1}{n})^n$. In the limiting case, if the interest is compounded continuously, the maximum amount $A$ that can be obtained at the end of one year is given by
This behaviour of applying an extremely small change an extremely large number of times occurs in many natural phenomena such as charging of a capacitor, decay of radioactive isotopes etc. In such processes, the small change is generally proportional to