We develop hyperasymptotic expansions for two general classes of homogeneous differential equations with an irregular singularity of arbitrary integer rank at infinity. The first is the general second-order differential equation. The second extends this result to a restricted case of a higher order linear differential equation. In both of these cases the result is expressed in terms of certain hyperterminant integrals which are generalisations of those found in other papers in the case of a rank one singularity.