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We introduce a model of technological advances as allowing for greater productivity at the cost of increased complexity. Complex goods and services require a large number of strongly complementary inputs. Innovation increases the probability that an input will be flawed, leading to a skewed distribution of firm sizes, with more extreme superstars and a higher share of mediocre firms. Whether additional combinatoric innovation increases growth is determined by a trade-off between its contribution to productivity and effective complexity. We evaluate strategies to deal with complexity, including modularization, and show they are just as important for boosting long-term growth as innovation.