Intuitive Understanding of the constant "$e$" Potentially related-questions, shown before posting, didn't have anything like this, so I apologize in advance if this is a duplicate.


I know there are many ways of calculating (or should I say "ending up at") the constant e.  How would you explain e concisely?
It's a rather beautiful number, but when friends have asked me "what is e?" I'm usually at a loss for words -- I always figured the math explains it, but I would really like to know how others conceptualize it, especially in common-language (say, "English").  

related but not the same: Could you explain why $\frac{d}{dx} e^x = e^x$ "intuitively"?

The thing that is special is the exponential function $\mathrm{exp}$, which satisfies
$$\mathrm{exp}(0)=1,\quad \mathrm{exp'}=\mathrm{exp}.$$
Then of course $e:=\mathrm{exp}(1)$, and because of $\mathrm{exp}(x+y)=\mathrm{exp}(x)\mathrm{exp}(y)$ it makes sense to write $\exp(x)=e^x$.
