We explore the paradigm in which inflation is driven by a four-dimensional strongly coupled dynamic with a nonminimal coupling to gravity. We introduce a model where the inflaton is identified with the glueball field of a pure Yang-Mills theory. We introduce the dilatonic-like glueball action, which is obtained by requiring saturation of the underlying Yang-Mills trace anomaly at the effective action level. We couple the resulting action nonminimally to gravity. We demonstrate that it is possible to achieve successful inflation with the confining scale of the underlying Yang-Mills theory naturally of the order of the grand unified energy scale. We also argue that the metric formulation gives a more consistent picture for models of composite inflation than the Palatini one. Finally, we show that, within the metric formulation, the model nicely respects tree-level unitarity for the scattering of the inflaton field all the way to the Planck scale.