I know percentages can be multiplied, as they're basically just fractions, so it makes sense to ask what 50% of 72% of 10 is, for example. But would anybody use an expression like 3%% as shorthand for 3/10,000? And so forth with as many '%' signs as you like (although with a lot it would be simpler to use scientific notation).
So, is this:
- A thing that's used and I've missed
- A thing that's syntactically correct but nobody uses it because there are better alternatives (which?)
- A thing that doesn't even make mathematical sense, and if so, could you help me understand why?
EDIT: I hadn't heard of per mille and basis point, thanks. Would it make sense, then, to put more little zeroes after the slash, one for each extra 10^(-1), continuing the pattern of percent, per-mille, and basis point?
This wiki article covers the issue with notation: there exist notions as "pro mille" and "basis point" ("one to thousand" and "one to ten thousand", respectively ).
So, to answer your three points:
your notation is - to my knowledge - not used
there're better notations: ‰ and ‱
if we translate $\%$ as the correposponding phrase ("one per hundred"), then $\%\%$ is not a valid phrase