  We prove a law of the iterated logarithm (LIL) for an infinite sum of
independent indicators parameterized by $t$ as $t\to\infty$. It is shown that
if the expectation $b$ and the variance $a$ of the sum are comparable, then the
normalization in the LIL includes the iterated logarithm of $a$. If the
expectation grows faster than the variance, while the ratio $\log b/\log a$
remains bounded, then the normalization in the LIL includes the single
logarithm of $a$ (so that the LIL becomes a law of the single logarithm).
Applications of our result are given to the number of points of the infinite
Ginibre point process in a disk and the number of occupied boxes and related
quantities in Karlin's occupancy scheme.
