  We construct, for any given positive integer $n$, Reeb flows on contact
integral homology 3-spheres which do not admit global surfaces of section with
fewer than $n$ boundary components. We use a connected sum operation for open
books to construct such systems. We prove that this property is stable with
respect to $C^{4+\epsilon}$-small perturbations of the Hamiltonian given on the
symplectization.
