Every (closed, connected, oriented) three-manifold contains a fibred knot. That is, every three-manifold has an open book decomposition with non-empty, connected binding and connected, oriented, compact page. This is due to Winkelnkemper or Gonzalez-Acuna - see page 625 of this book.
So, even your example of a product (of a closed surface with a circle) has some non-trivial open book decomposition. It just is not the "obvious" one.