  We bound from below the complexity of the top Chern class of the Hodge bundle
in the Chow ring of the moduli space of curves: no formulas in terms of classes
of degrees 1 and 2 can exist. As a consequence of the Torelli map, the
0-section over the second Voronoi compactification of the moduli of principally
polarized abelian varieties also can not be expressed in terms of classes of
degree 1 and 2. Along the way, we establish new cases of Pixton's conjecture
for tautological relations.
  In the log Chow ring of the moduli space of curves, however, we prove that
the top Chern class of the Hodge bundle lies in the subalgebra generated by
logarithmic boundary divisors. The proof is effective and uses Pixton's double
ramification cycle formula together with a foundational study of the
tautological ring defined by a normal crossings divisor. The results open the
door to the search for simpler formulas for the top Chern class of the Hodge
bundle on the moduli of curves after log blow-ups.
