jmc

A real number $x$ is in the domain of $$f(x)=\frac{1}{x-64}+\frac{1}{x^2-64}+\frac{1}{x^3-64}$$ unless at least one of $x-64$, $x^2-64$, $x^3-64$ is equal to $0$. This occurs for $x=64$, $x=8$, $x=-8$, and $x=4$, so there are $\boxed{4}$ real numbers not in the domain of $f$.