The electric permittivity and the refractive index are fundamental properties of electromagnetic or optical materials. Their values are determined by various electromagnetic resonances present in the system. While it is possible to obtain extreme values of effective material properties by designing artificial resonances originating from the shapes and sizes of mesoscopic inclusions, often the resulting properties become strongly dispersive. Here, we look at the ways to overcome this and achieve unprecedented values of permittivities and refractive indices, which are nearly dispersion-less over a broad frequency range. This broadband manipulation of effective material properties can be particularly useful in applications in which a large bandwidth or large fabrication error tolerance is desired.