Some contact manifolds have the property that the Reeb flow of every contact form supporting the corresponding contact structure has positive topological entropy. Examples of such manifolds are given by the unit sphere bundle of rationally hyperbolic manifolds and manifolds whose fundamental group has exponential growth. In this talk I will discuss the construction of new examples using contact connected sums. If time allows, I will briefly explain how the idea of this construction leads us to the development of a Lagrangian Floer homology on the complement of certain codimension two invariantsubmanifolds in cotangent bundles. This is joint work with Marcelo Alves.