Because the call:
R = ones(100,100);
rank(R,1e-20)
attempts to do something that is demonstrably silly, in terms of double precision arithmetic.
You can try to squeeze blood out of a rock, but when you do, the only blood you will see comes from your own fingers. The default tolerance in rank is set reasonably to produce a valid result in double precision arithmetic. Making it significantly smaller does not make the code work better. Instead, you are fooling yourself.
What does rank do with the tolerance you give it? It essentially counts to see how many singular values of the matrix are sufficiently non-zero, when compared to the size of the maximum singular value.
S = svd(R);
The default tolerance for rank is:
defTol = max(size(R)) * eps(max(S))
As you can see, it uses that maximum singular value, to then produce a reasonable tolerance for this problem.
sum(S > defTol)
But by setting the tolerance to a far too small number, you create a situation where rank returns something meaningless.
