Is there a simple geometrical description of $e$? Of course I am not looking for a definition through $\int_1^e{1\over x} \, \mathrm{d}x=1$ or that slope of $a^x$ at $x=0$ is $1$ when $a=e$.  I am looking for something understandable by a kid who has begun comprehending $\pi$ as the ratio of circumference to diameter of a circle. Or perhaps by one who is a couple of years older.
(And of course there is no reason to suspect that every mathematical constant has a simple geometrical  description. But a "definition" that might not be suitable for a calculus text could be suitable for introduction to laymen.)
Edit 1:
One area I would find interesting would be a definition that uses the entire hyperbola $xy=1$ instead of pieces of it. Of course both area (between hyperbola and asymptotes) and the length, measured in the usual fashion, will be infinity.  I have attempted projecting the shape onto a sphere to see if I get a number similar to $e$; but with no luck.
 A: Math is fun has a nice description:  http://www.mathsisfun.com/numbers/e-eulers-number.html
If you divide up a number into $n$ parts and multiply them together, the answer is biggest when your number is cut up to a value near $e$.  It represents the best sized chunks of a number to multiply together.
A: I am not sure if this is "geometric", but I find it intuitive. A definition of $e$ is the following limit, $$lim_{x\to 0}(1+\frac{1}{n})^n.$$ At this point you may be saying "this is not very intuitive" and at this point, you would be right. So  let me explain what is intuitive about this definition. The intuition comes from contemplating compound interest. If your interest rate is $r$, your future value is $A$, your present value is $P$, the number of years you let your account sit $t$, and the number of compounds per year is $m$, then the future value is given by, $$A=P(1+\frac{r}{m})^{mt}.$$ 
Now if invest one dollar at the beginning of the year at a rate of 100%, (we will let the number of compounds per year vary) then we get $$A=(1+\frac{1}{m})^{m}.$$ Now what if we allow our interest to compound continuously? Then in some sense we have an infnite number of compounds per year, so we take the limit as $m$ goes to infinity and we will have $e$ dollars at the end of the year. 
Now this may seem sort of silly, but if we want to modify our compound interest equation to deal with continuously compounding interest, then the formula that we obtain is $$A=Pe^{rt}.$$ This turns out to be a useful formula for a number of reasons. If we have interest that is compounding by the second, then continuously compounded interest is a good approximation. This may come in handy when various financial firms are making many computerized trades per second (called high frequency trading).  
A: In fact your first definition $\ \displaystyle\int_1^e \frac {dx}x\ $ allows a nice geometrical definition :

*

*Draw the hyperbola $\ x\mapsto \frac 1x$


*Represent the first square $[0,1]\times[0,1]\,$ and write $[1]$ inside it


*Write $1$ at the right and top of this square


*Write $\,1, e, e^2, e^3\cdots\ $ near the $x$ and $y$ axis so that the areas (delimited by two vertical and two horizontal lines) are $1$ every time.
You could try too some arithmetic : let the students compute $(1+1)^1$, $(1+1/2)^2$, $(1+1/3)^3\cdots$
Ask them if there is a limit. Explain that the limit is not a rational number...
