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][1].  

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. 

  [1]: https://www.maths.ed.ac.uk/~v1ranick/books/knot.pdf