We demonstrate the existence of a novel quasiparticle, an exciton in a semiconductor doubly dressed with two photons of different wavelengths: a near infrared cavity photon and terahertz (THz) photon, with the THz coupling strength approaching the ultrastrong coupling regime. This quasiparticle is composed of three different bosons, being a mixture of a matter-light quasiparticle. Our observations are confirmed by a detailed theoretical analysis, treating quantum mechanically all three bosonic fields. The doubly dressed quasiparticles retain the bosonic nature of their constituents, but their internal quantum structure strongly depends on the intensity of the applied terahertz field.