  We study almost complete intersections ideals whose Rees algebras are
extremal in the sense that some of their fundamental metrics---depth or
relation type---have maximal or minimal values in the class. The focus is on
those ideals that lead to almost Cohen--Macaulay algebras and our treatment is
wholly concentrated on the nonlinear relations of the algebras. Several classes
of such algebras are presented, some of a combinatorial origin. We offer a
different prism to look at these questions with accompanying techniques. The
main results are effective methods to calculate the invariants of these
algebras.
