Question: What would the last digit be in a base 11 system? Base 12?

I'm mostly wondering because I was thinking of a base 36 system with digits 0-9 immediately followed by the letters A-Z but I wasn't sure if that would be considered a proper numeral system, or if there was something else in place.

Answer: 
The extra digits can be anything you want (the symbols 0-9 themselves are arbitrary, and could be switched around or replaced by other things, though that'd be adding unnecessary confusion). As Arturo mentions above, [hexadecimal notation](http://en.wikipedia.org/wiki/Positional_notation#Computing) is common in computing, and there it is conventional to use $A=10$ through $F=15$. 

If you want to stick to using the 10 [Hindu-Arabic numerals](http://en.wikipedia.org/wiki/Hindu%E2%80%93Arabic_numeral_system) and 26 English letters, and want to go past base 36, the [Wikipedia article on positional notation](http://en.wikipedia.org/wiki/Positional_notation#Digits_and_numerals) describes how one might solve this by using groups of these symbols in each position.
IIRC, on the SWAC at UCLA in the 50's and 60's, the 6 hex digits beyond 9 were U V W X Y Z.

Also, for some reason, this pops into my mind for the hex digits: G B N J F L. These stood for "Great Big Numbers Just For Laughs".