$e$ is often explained in terms of compound interest. If I found a bank that gave me 100% annual compound interest, then if I put in £1.00, at the end of the year, I would have £2.00. If I were more savvy, and instead asked for 50% interest paid biannually, then I would end up with more – £2.25 to be exact. (This is because 50% of £1.50 > 50% of £1 – simple interest, rather than compound interest, would still only give me £2.00.) $e$ appears to be the logical extreme of this idea: of taking $\frac{100%}{n}$% interest $n$ times per year. I understand it as the limit of $(1+1/n)$ as $n$ tends to infinity. When the analogy starts to break down for me is when it is therefore concluded that you can take the interest infinitely/continually often. Obviously, this is conceptually harder already, because of the introduction of infinity. However, it is also seems to beg the question "what is the interest rate?". If it is 0%, then the £1.00 will never increase, but any more than 0%, and then the individual interest rates would no longer add up to 100%. Is it some kind of infinitesimal?
To illustrate my wariness, I have this example from the wikipedia article on limits (https://en.wikipedia.org/wiki/Limit_(mathematics)): $$f(x)=\frac{x^2-1}{x-1}$$ As $x$ gets arbitrarily close to 1, $f(x)$ approaches 2, no matter which side you approach 1 from. However, $f(1)$ is undefined as involves division by zero. Similarly, as $n$ tends to infinity in the $e$ analogy, the growth rate becomes arbitrarily close to $e$. But I don't see how this means that when $n=\infty$, the growth rate is necessarily $e$. After all, if you plug $n=\infty$ into the normal formula $(1+1/n)^n$, it seems that it breaks down (forgive me if you cannot use infinity in this way).
e is defined (or one of its definitions) to be $$\lim _{n \to \infty} (1+\frac{1}{n})^n$$, and not to be when you plug in n as infinity. Most of our math is defined as a limit, and doesn’t make any sense when you simply plug in infinity. For example, take the simple function $\frac{1}{x}$. You can say the limit as x goes to infinity is zero, but you cannot say that $\frac{1}{\infty}$ is zero, because our basic functions are only defined for finite numbers. When we say interest taken continually we mean interest taken as defined by a limit to infinity.