Why are there letters as additional digits in bases greater than the decimal base (10)? For example, in the hexadecimal base (16), the letters A through F are included as digits, but why?  This letter thing happens in bases greater than ten.  For example, in the hexadecimal base, C is 12 in the decimal base and E is 14 in the decimal base.  Also, in base 20, the letters A through J are included as digits.  Why are there letters for extra digits in bases greater than ten?  I'm thinking your answers will be "positive!"
 A: Well you have to use some symbols because the conventionally used digits only go up to $9$. You could just invent new symbols, but then typesetters, font makers and students learning the notation would curse your name for decades and centuries after you die. It's better to just use what we already have (although I myself don't like this that much).
A: Why are there these crazy symbols $\Large \;\mathbf{2}$, $\;\Large \mathsf{3}$, $\Large \;\mathit{4}$, ... $\Large\;\mathtt{9}$ in bases greater than base 10?
("10" of course denoting the number of eyeballs most people have.)

The point is, how would you explain these symbols to someone who had only ever heard of binary representation? Here's what I would say. 

I know you've only ever used $0$ and $1$ before, but long ago some people agreed on arbitrary additional symbols to represent the numbers $10$ (the number of dots in ${:}$) through $1001$ (the number of dots in ${:}{:}{:}{:}{\cdot}$), and in fact you can make a perfectly fine numeral system where instead of counting
  $$0, \quad 1,\quad 10,\quad 11,\quad 101,\quad\ldots$$
  like normal, you instead continue by using these arbitrary symbols
  $$0,\quad 1,\quad 2,\quad 3,\quad 4,\quad 5,\quad\ldots$$
  and only write the symbol $10$ when you run out of extra symbols. The place where you run out is called the base of the numeral system.

A: Of course you could write the number 65535 (FFFF in conventional base-16 representation) as $<15; 15; 15; 15>_{16}$ or something like that; it's clear, it's general-purpose, and it works beyond base-36, but it's just less convenient.
A: We need to distinguish between the symbol $10$, which means the number $1\cdot b + 0\cdot b^0$ in base $b$, and the number which we designate by $10$ in our normal arithmetic, by which I mean the tenth positive integer.
In hexadecimal, $10$ should mean $1\cdot 16 + 0\cdot 16^0$, which is the sixteenth number. If you want to refer to the tenth integer, you need to give it a name. We give it the name $A$, but really it just needs any name and we are only so creative, it seems.
A: When Fibonacci introduced the Indo-Arabic system in his Liber Abaci, he draw the digits inspired from the shapes used by the Arabs. At that time they were just weird symbols and it took time for people to get used to them.
It has then become customary use a limited set of them for the binary system or systems with base less than ten (I write it in letters on purpose). There's no particular reason, one could decide that @ and ! are the symbols for zero and one in the binary system: any two symbols would be as good. Using $0$ and $1$ is better because we need no “translation”.
For the hexadecimal system we need sixteen distinct symbols; we already have ten for denoting the numbers from zero to nine, so it's natural to use them and it remains to choose other six. Letters have an established ordering, and this is the simple reason why they were chosen.
The Babylonians used a base sixty system, so they needed as much distinct symbols. Their choice was to make up them with two base symbols (for one and ten) grouping them in a clever way

For zero the symbol was

A similar system was used by the Mayas who used base twenty:

Note that both systems were quite good for addition: in the Mayan system, five dots became a bar and four bars implied carrying a dot to the next column.
As you see, there's no need of using letters, it's just a convenience.
A: Teaching kids about the idea that you can represent numbers in bases other than the 10 we grew up with raises really interesting philosophical questions about distinguishing between a number and the name we give that number. Once they realize that you need as many digits as the base and that the next number will be written as "10" you have to struggle to remind them not to read that out loud as "ten".
I know you're not supposed to provide answers as links, but if you want to read more about some of my experience along these lines, try
http://www.cs.umb.edu/~eb/sam/duodecimal/ssegm.pdf .
