How would you count a base $> 36$ system?

You can choose your favorite letter and put an index, e.g. for base $n\in \mathbb{N}$ $$a_1,a_2,a_3,\ldots, a_n.$$

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.

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.