We devise a multi-scale Hybrid High-Order (HHO) method. The method hinges on (hybrid) discrete unknowns that are polynomials attached to mesh elements and faces, and on a multi-scale reconstruction operator, that maps onto a fine-scale space spanned by oscillatory basis functions. The method handles arbitrary orders of approximation k >= 0, and is applicable on general meshes. For face-based unknowns that are polynomials of degree k, we devise two versions of the method, depending on the polynomial degree (k - 1) or k of cell-based unknowns. We prove, in the case of periodic coefficients, an energy-error estimate of the form (ε^(1/2) + H^(k+1) + ε^(1/2)H^(-1/2)).