  We use contact fiber sums of open book decompositions to define an infinite
hierarchy of filling obstructions for contact 3-manifolds, called planar
k-torsion for nonnegative integers k, all of which cause the contact invariant
in Embedded Contact Homology to vanish. Planar 0-torsion is equivalent to
overtwistedness, while every contact manifold with Giroux torsion also has
planar 1-torsion, and we give examples of contact manifolds that have planar
k-torsion for any $k \ge 2$ but no Giroux torsion, leading to many new examples
of nonfillable contact manifolds. We show also that the complement of the
binding of a supporting open book never has planar torsion. The technical basis
of these results is an existence and uniqueness theorem for J-holomorphic
curves with positive ends approaching the (possibly blown up) binding of an
ensemble of open book decompositions.
