  We compute for all orientable irreducible geometric 3-manifolds certain
complexity functions that approximate from above Matveev's natural complexity,
known to be equal to the minimal number of tetrahedra in a triangulation. We
can show that the upper bounds on Matveev's complexity implied by our
computations are sharp for thousands of manifolds, and we conjecture they are
for infinitely many, including all Seifert manifolds. Our computations and
estimates apply to all the Dehn fillings of M6_1^3 (the complement of the
three-component chain-link, conjectured to be the smallest triply cusped
hyperbolic manifold), whence to infinitely many among the smallest closed
hyperbolic manifolds. Our computations are based on the machinery of the
decomposition into `bricks' of irreducible manifolds, developed in a previous
paper. As an application of our results we completely describe the geometry of
all 3-manifolds of complexity up to 9.
