Summary: We consider sixth-order theories of gravity with Lagrangians containing the term \(R^ N\square R\), where \(N > 1\) is an integer. A conformal transformation gives a picture with two scalar fields in Einstein’s theory. We find de Sitter and power law inflationary solutions. However, the theory has an unstable weak-field behaviour, because the linearized field equation is of lower order than the non-linear one.