Using log^2(x) to denote log(log(x)) is far from nonsensical. It makes sense when keeping in mind the fact that function composition is an operation on functions much like addition and multiplication. For example, we can write
f(f(x)) = (f∘f)(x) = f^2(x)
f(f(f(x))) = (f∘(f∘f))(x) = (f∘f∘f)(x)
an so on. We can see that when composing a function with itself many times it is much easier to write f^n(x) than the alternative.
Using log^2(x) to mean (log(x))^2 is a shortcut that typesetters took in order to save time and is not as widely used in mathematics.