In this paper, a hyperchaotic memristive circuit that is based on a Wien bridge oscillator structure is introduced. The circuit contains an active flux-controlled memristor that is characterized with a smooth continuous cubic nonlinearity. Firstly, a floating implementation of the memristor is used in order to show the flexibility of the element and consequently, a typical application example is analyzed by employing it as the nonlinear element in Chua circuit. Later, the simulation and the experimental realization of the hyperchaotic memristive circuit are performed, resulting in a consistent match between them. Moreover, the dynamics of the proposed hyperchaotic system are theoretically examined in detail. When compared to similar work in the literature, the proposed hyperchaotic system demonstrates clear improvements such as non-clipping amplitude behavior and simplified circuit implementation along with rich chaotic dynamics. The proposed circuit can further be implemented for chaotic encryption, communication systems and synchronous control applications.