The nonlinear oscillations of acoustically forced spherical gas bubbles in an upper-convected Maxwell (UCM) compressible fluid are investigated. The nonlinear viscoelastic model used is suitable for large-amplitude excitation of bubbles that cannot be captured by linear models. The effects of acoustic excitation are studied for compressible nonlinear viscoelastic media, which increases the complexity and nonlinearity of the behavior. The Keller-Miksis equation is used to model the dynamics of a single bubble. The constitutive equations of compressible UCM are used for viscoelastic media. These governing equations are non-dimensionalized and coupled to determine the bubble dynamic behavior. The set of derived non-dimensionalized integro-differential equations developed are numerically solved simultaneously using the fourth-order Runge-Kutta method featured by the automatic variable time step-size module. The combined effects of compressibility and viscoelasticity of the fluid on bubble radius are investigated. The results show that the combination of compressibility and nonlinear viscoelasticity for bubble radial oscillations makes forced bubble dynamics more applicable for human needs, especially for large deformations in highly non-Newtonian fluids like industrial polymers or even tissue-like media. It can be seen that compressibility controls the oscillations at higher forcing amplitudes. The relevance and importance of these bubble dynamics to biomedical ultrasound applications and light emissions by sonoluminescence and other industries are evident.