Always oddly-many ones in the binary expression for $10^{10^{n}}$?
The answer to the title question is "no" according to the following computation in a Sage notebook:
%time
for n in [1..10]: print n, (5^10^n).popcount()
1 11
2 105
3 1163
4 11683
5 115979
6 1161413
7 11606847
8 116093517
9 1160951533
10 11609679812
CPU time: 487.60 s, Wall time: 1935.41 s
EDIT: In Sage, the popcount() method -- which returns the number of ones in the binary representation -- is built-in for objects of type 'Integer' (but not for type 'long', which I had been forcing), making it unnecessary to import gmpy, etc., as done previously.