  In the Karlin infinite occupancy scheme, balls are thrown independently into
an infinite array of boxes $1$, $2,\ldots$, with probability $p_k$ of hitting
the box $k$. For $j,n\in\mathbb{N}$, denote by $\mathcal{K}^*_j(n)$ the number
of boxes containing exactly $j$ balls provided that $n$ balls have been thrown.
We call $\textit{small counts}$ the variables $\mathcal{K}^*_j(n)$, with $j$
fixed. Our main result is a law of the iterated logarithm (LIL) for the small
counts as the number of balls thrown becomes large. Its proof exploits a
Poissonization technique and is based on a new LIL for infinite sums of
independent indicators $\sum_{k\geq 1}\Bbb{1}_{A_k(t)}$ as $t\to\infty$, where
the family of events $(A_k(t))_{t\geq 0}$ is not necessarily monotone in $t$.
The latter LIL is an extension of a LIL obtained recently by Buraczewski,
Iksanov and Kotelnikova (2023+) in the situation that $(A_k(t))_{t\geq 0}$
forms a nondecreasing family of events.
