  A law of the iterated logarithm is established for the last passage times of
directed percolation on rectangles in the plane over exponential or geometric
independent random variables, rescaled to converge to the Tracy-Widom
distribution. The normalization is of loglog type, in contrast with the log
normalization for the largest eigenvalue of a GUE matrix recently put forward
by E. Paquette and O. Zeitouni. The proof relies on sharp tail bounds and
superadditivity, close to the standard law of the iterated logarithm. A weaker
result for the liminf is also discussed.
