  Kifer, Peres, and Weiss proved that there exists $c_0>0,$ such that $\dim
\mu\leq 1-c_0$ for any probability measure $\mu$ which makes the digits of the
continued fraction expansion i.i.d. random variables. In this paper we prove
that amongst this class of measures, there exists one whose dimension is
maximal. Our results also apply in the more general setting of countable
branched systems.
