Otto's Theorem characterises the bisimulation-invariant PTIME queries over graphs as exactly those that can be formulated in the polyadic µ-calculus, hinging on the Immerman-Vardi Theorem which characterises PTIME (over ordered structures) by First-Order Logic with least fixpoints. This connection has been extended to characterise bisimulation-invariant EXPTIME by an extension of the polyadic µ-calculus with functions on predicates, making use of Immerman's characterisation of EXPTIME by Second-Order Logic with least fixpoints.In this paper we show that the bisimulation-invariant versions of all classes in the exponential time hierarchy have logical counterparts which arise as extensions of the polyadic µ-calculus by higher-order functions. This makes use of the characterisation of k-EXPTIME by Higher-Order Logic (of order k + 1) with least fixpoints, due to Freire and Martins.