  We establish a law of the iterated logarithm (LIL) for the set of real
numbers whose $n$-th partial quotient is bigger than $\alpha_n$, where
$(\alpha_n)$ is a sequence such that $\sum 1/\alpha_n$ is finite. This set is
shown to have Hausdorff dimension $1/2$ in many cases and the measure in LIL is
absolutely continuous to the Hausdorff measure. The result is obtained as an
application of a strong invariance principle for unbounded observables on the
limit set of a sequential iterated function system.
