Two recent theoretical studies of adaptation suggest that more complex organisms tend to adapt more slowly. Specifically, in Fisher's "geometric" model of a finite population where multiple traits are under optimizing selection, the average progress ensuing from a single mutation decreases as the number of traits increases--the "cost of complexity." Here, I draw on molecular and histological data to assess the extent to which on a large phylogenetic scale, this predicted decrease in the rate of adaptation per mutation is mitigated by an increase in the number of mutations per generation as complexity increases. As an index of complexity for multicellular organisms, I use the number of visibly distinct types of cell in the body. Mutation rate is the product of mutational target size and population mutation rate per unit target. Despite much scatter, genome size appears to be positively correlated with complexity (as indexed by cell-type number), which along with other considerations suggests that mutational target size tends to increase with complexity. In contrast, effective population mutation rate per unit target appears to be negatively correlated with complexity. The net result is that mutation rate probably does tend to increase with complexity, although probably not fast enough to eliminate the cost of complexity.