Pushing the Iterated Logarithm Rule to the limits.

Question

Given a standard Brownian motion, the Iterated Logarithm Rule says that with probability one, $$\frac{|w(t)|}{\sqrt{t \log\log t}},\ (1\le t)$$ has $\limsup$ $\sqrt{2}$ as $t \to\infty$. But what is the chance of $$\frac{|w(t)|}{\sqrt{t \log\log t}}=\sqrt{2}+1/\sqrt{t}$$ ever happening for at least one $t$, ($1\le t$)? (Note that it does not follow from the above rule that it is a sure thing.) If you can show it is in (0,1), I would be alreadey happy.