Noting that the largest eigenvalue of a random matrix and the maximal increasing subsequence of a random permutation are both expressed by the Tracy-Widom distribution, I might broaden your question to appearances of the law of iterated logarithm in connection with Tracy-Widom. The application I am aware of involves the self-interacting one-dimensional stochastic process called "true self-repelling motion". The law of iterated logarithm of the position $X_t$ at time $t$ for this process was derived by Laure Dumaz in <A HREF="http://arxiv.org/abs/1105.2948">Large deviations and path properties of the true self-repelling motion</A>: *almost surely,*

$$\lim {\rm sup}_{t\rightarrow\infty}\frac{X_t}{t^{2/3}(\ln\ln t)^{1/3}}=2^{-2/3}\frac{3}{|a_1|},$$

with $a_1$ the first (negative) zero of the derivative of the Airy function.

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What strikes me, when comparing with the usual scaling for Brownian motion, is that the factor $t$ in the denominator and the iterated logarithm appear with different powers. Simply looking at the different scaling of the root-mean-square displacement ($t^{1/3}$ rather than $t^{1/2}$), I would have expected the same power $1/3$ for both. I would guess an intuitive explanation must be quite subtle.