We present a tight bound on the exact maximum complexity of Minkowski sums of convex polyhedra in $$³. In particular, we prove that the maximum number of facets of the Minkowski sum of two convex polyhedra with $m$ and $n$ facets respectively is bounded from above by $f(m,n)\; =\; 4mn\; -\; 9m\; -\; 9n\; +\; 26$. Given two positive integers $m$ and $n$, we describe how to construct two convex polyhedra with $m$ and $n$ facets respectively, such that the number of facets of their Minkowski sum is exactly $f(m,n)$. We generalize the construction to yield a lower bound on the maximum complexity of Minkowski sums of many convex polyhedra in ³. That is, given $k$ positive integers $m\_1,m\_2,...,m\_k$, we describe how to construct $k$ convex polyhedra with corresponding number of facets, such that the number of facets of their Minkowski sum is exactly $\sum$_{1 i < j k}(2m_i - 5)(2m_j - 5) + binom(k,2). We also provide a conservative upper bound for the general case.