In this paper, we extend Landau's notion of `exchange relations' so as to make sense for arbitrary planar algebras, which need not necessarily be generated by its `$2$-boxes'. We show, as in Landau's case, that these `higher exchange relation planar algebras' are necessarily `finite dimensional', and that examples of such planar algebras are given by all (even possibly reducible) depth two subfactors, as well as planar algebras associated to subfactors with principal graphs $E_6$ and $E_8$.