This paper is concerned with the integrodifferential Benjamin-Bona-Mahony equation u(t) - u(txx) + alpha u - integral(infinity)(0)g(s)u(xx)(t - s)ds + (f(u))(x) = h complemented with Dirichlet boundary conditions, in the presence of a possibly large external force h. The nonlinearity f is allowed to exhibit a superquadratic growth, and the dissipation is due to the simultaneous interaction between the nonlocal memory term and the Rayleigh friction. The existence of regular global and exponential attractors of finite fractal dimension is shown.