  We consider the GUE minor process, where a sequence of GUE matrices is drawn
from the corner of a doubly infinite array of i.i.d. standard normal variables
subject to the symmetry constraint. From each matrix, we take its largest
eigenvalue, appropriately rescaled to converge to the standard Tracy-Widom
distribution. We show the analogue of the law of iterated logarithm for this
sequence, i.e. we divide the normalized n-th eigenvalue by a logarithmic factor
and show the limsup of this sequence is a constant almost surely. We also give
almost sure bounds for the appropriately scaled liminf.
