How would you count a base $> 36$ system? When I am counting in a base greater than ten, I can use the letters of the alphabet.  What do I use when I run out of those?
What comes after $z$?:
$0, 1, 2, 3,\ldots, 9, a, b, c, d,\ldots, x, y, z$ (?)
And how would I find it?
 A: We have tons of symbols available, Take any number of languages which have pairways disjoint letters. Index each of those languages and use the pre-ezisting indexes inside the language and you shall have a lot of letters, just by using the symbols for base $10$ numbers ,Japanese Hiragana, katakana, Arab, Hebrew and Spanish we get $179$ symbols.
A: +, -, /, { ,], ä, ö, ü, ß, ă, î, ț, ﺞ, 덖, 哶, ...
There are millions of characters you could use. There are standards mostly used for base $64$  or base $32$ but I'd say that it's mostly up to you what you choose.
A: I don't mean to be silly, but why not use base 6?  If you want join the gitis in groups of two bits
1234123051 can be understood as a 5-digit number:
12 34 12 30 51
In the decimal system multiplying by ten $T: x\mapsto 10x$ plays a very special role.  
If we want to learn about a number $\alpha$  we could take the integer part (e.g. $\lfloor 3.5\rfloor = 3$), then multiply by 10, take the integer part again.  Doing this over and over would give our decimal system.


*

*$10^0\pi \mapsto 3$

*$10^1\pi \mapsto 1$

*$10^2\pi \mapsto 4$

*...


In your case, using 36 instead of 10.  Or just use two base-6 digits.  This is also related to channel coding from information theory.
A: You can choose your favorite letter and put an index, e.g. for base $n\in \mathbb{N}$
$$a_1,a_2,a_3,\ldots, a_n.$$
A: The Babylonians were able to do math in base $60$. Each digit consisted of $N$ wedge marks in one direction and $M$ in the other direction, and the value of the digit was $10N + M$. They lacked a zero, but that is not hard to provide.
You could also say that time of day is base $60$, so 
$\mbox{1:23:45} = 1 \cdot 60^2 + 23 \cdot 60 + 45.$
The basic idea of a place-value system is that you can tell what digit is in each place
somehow (in this case, by taking the symbols $0$ through $9$ in pairs, and some people insert the symbol ":" beteween pairs as a visual aid).
Each digit does not need to be drawn as a single connected region of black paint.
A: Take for example base $64$, widely used for encoding binary data.
Once it gets to Z (it starts uppercase) it goes with lowercase letters: WXYZabcd. The careful observer will note it's still missing two letters, because $10 + 2 \cdot 26 = 62.$
Those two are usually one of /, + and -, but it's more or less implementation dependent.
A base $64$ number looks like this cGxlYXN1cmUu and you have probably seen it in a URL before.
A: You can also encode each digit of high-base number system by fixed-width representation of a number in low-base number system. For example, each digit of a number in base $\leq 256$ can be encoded by a pair of hexadecimal digits.
