I read this page
http://mathworld.wolfram.com/e.html
But every method to calculate e always begins by generating increasing accuracy from left to right. Is it possible to calculate the 10^9th digit only? Or is it possible to calculate the range of 10 digits from the 10^15th place to the 10^16th? Obviously one could spend a lot of cpu cycles to get to the target digit and only display that one but that is not what I am asking. Rather is it possible to have an expression that generates the digit of interest directly?
Such an algorithm can't be ruled out, but currently there is no known method to generate the $n$th decimal digit of $e$ without generating the preceding digits first.
Remarkably, there is an algorithm for generating the $n$th hexadecimal digit of $\pi$ directly. This is the Bailey–Borwein–Plouffe formula discovered by Simon Plouffe in 1995.