Given a standard random walk, the Iterated logarithm rule say that with probability one, $$\frac{|w(n)|}{\sqrt{n \log\log n}}$$ has $\limsup$ $\sqrt{2}$ as $n \to\infty$.
What about other values? What is the chance that $\limsup= c$, where $c$ is bigger than $\sqrt{2}$? In particular, for what $c$ value the chance will be $\frac{1}{2}$?
The random variable $K=\limsup_{n\to\infty} w(n)/\sqrt{n\log\log n}$ can, for some sample paths, be bigger or smaller than $\sqrt 2$. With probability $1$, however, it equals $\sqrt 2$, and hence the probability that it is less than $\sqrt 2$ is zero, and so is the probability that it is greater than $\sqrt 2$.
You might be confusing the shape of the law of the iterated logarithm, where the probability is "outside" the limiting operation, with that of the central limit theorem, where one has a limit of probabilities, that is, the limit is "outside" and the probability "inside". From this point of view your question is a bit like asking, in the strong law of large numbers, we know that the sample average $\bar X_n\to\mu$ with probability one, and you ask for the chance that $\bar X_n\to\mu+1/10$. The answer is : zero, because the $\mu$ limit has gobbled up all the probability in town.