A common question is why e (2.71828...) is so special. Why not 2, 3.7 or some other number as the base of growth?
First off, e was discovered, not chosen. Think of the speed of light, c. It wasn't originally decided to be 299,792,458 m/s  we did experiments and realized under ideal, universal conditions (a vacuum), this was the fastest light could move^{1}.
Let's look at growth and ask under ideal, universal conditions, what's the fastest something can possibly grow? Ideal, universal assumptions would be:
 Growing by the unit rate (100%)
 Growing for the unit time (1 time period)
 Growing perfectly, without any delay (continuous)
Turning these assumptions into a formula, we get:
If we actually use the formula (using large values of n for more accuracy) we get e = 2.718281828459...
Objection: But $13.74^x$ can model exponential growth just like $e^x$ can!
Sure. But what assumptions did you make to get 13.74? They probably weren't "unit rate, unit time, perfectly compounded". (You can pick k as the speed of light through Kool Aid too  but why?)
Arguably, $2^n$ is also universal ("the discrete e"), because you have zero compounding (n is an integer like 0, 1, 2, 3). Instead of perfectly continuous, it's perfectly noncontinuous (discrete), and we take growth stepbystep.
So, I'd say either $2^n$ (in discrete systems) or $e^x$ (in continuous systems) are "universal".
Objection: But things can grow faster than $e^x$, which is just $2.71828^x$  what about $13.74^x$?
What is it with you and 13.74? Yes, you can beat $e^x$ in an exponential footrace, if you use a rate more than 100%. $13.74^x$ is really $[e^{\ln(13.74)}]^x$. Because ln(13.74) ~ 2.6, you are assuming a 260% continuous interest rate, more than the 100% $e^x$ uses. (Alternatively, you can grow for 260% of the unit time period that $e^x$ uses.)
Related:

Funny enough, in 1983 c was decreed to be 299,792,458 m/s by redefining the length of a meter. Similarly, you could decide that e is a clean "10" in your basee number system. ↩
Other Posts In This Series
 An Intuitive Guide To Exponential Functions & e
 Demystifying the Natural Logarithm (ln)
 A Visual Guide to Simple, Compound and Continuous Interest Rates
 Common Definitions of e (Colorized)
 Understanding Exponents (Why does 0^0 = 1?)
 Using Logarithms in the Real World
 How To Think With Exponents And Logarithms
 Understanding Discrete vs. Continuous Growth
 What does an exponent really mean?
 Q: Why is e special? (2.718..., not 2, 3.7 or another number?)