  We consider pointwise, box, and Hausdorff dimensions of invariant measures
for circle diffeomorphisms. We discuss the cases of rational, Diophantine, and
Liouville rotation numbers. Our main result is that for any Liouville number
$\tau$ there exists a $C^\infty$ circle diffeomorphism with rotation number
$\tau$ such that the pointwise and box dimensions of its unique invariant
measure do not exist. Moreover, the lower pointwise and lower box dimensions
can equal any value $0\le \beta \le 1$.
