Can we ever know the sum of the first $10^{10^{100}}$ digits of $\pi$?

Can we calgulate the $n$th digit of $\pi$ when the Kolmogorov-complexity of $n$ is larger than the complexity of the calculating system?

Example: We cannot calculate digit number 314159265358 of $\pi$ on a typical pocket calculator.

EDIT: As Goldstern remarks, we cannot write down all the digits simultaneously. But can we know every desired shorter string of digits, for instance the first billion following the digit number $10^{{10}^{90}}$ and then note their sum, and so on?