We show that in integro-differential delayed dynamical systems, an hybrid state of simultaneous fast-scale hyperchaos and slow-scale periodicity can emerge subsequently to a sequence of Hopf bifurcations. The resulting timetrace thereby consists in hyperchaotic oscillations breathing periodically at a significantly lower frequency. We present experimental evidence of this counterintuitive type of dynamics in a Mach-Zehnder modulator optically fed by a semiconductor laser and subjected to nonlinear electro-optical feedback. We provide a theoretical understanding of this phenomenon, and we discuss its similarity with some dynamical behaviors observed in thermo-optical Fabry-Perot cavities and in neurons.