Geometric mean without overflowing the product

The geometric mean of $n$ positive numbers is the $n$-th root of their product, $\left(\prod_i x_i\right)^{1/n}$. It is the right average for growth rates and multiplicative returns. But the obvious code multiplies everything first, and a product of even a few dozen mid-sized numbers overflows a 64-bit integer or a double's range almost immediately.

Compute the geometric mean of a list of positive numbers without ever forming the full product.