Consider an infinite array of standard complex normal variables which are independent up to Hermitian symmetry.  The eigenvalues of the upper-left NxN submatrices, form what is called the GUE minor process.  This largest-eigenvalue process is a canonical example of the Airy process which is connected to many other growth processes.  We show that if one lets N vary over all natural numbers, then the sequence of largest eigenvalues satisfies a 'law of fractional logarithm,' in analogy with the classical law of iterated logarithm for simple random walk.  This GUE minor process is determinantal, and our proof relies on this.  However, we reduce the problem to correlation and decorrelation estimates that must be made about the largest eigenvalues of pairs of GUE matrices, which we hope is useful for other similar problems.