  We study the strong gravitational lensing in the background of a rotating
non-Kerr compact object with a deformed parameter $\epsilon$ and an unbound
rotation parameter $a$. We find that the photon sphere radius and the
deflection angle depend sharply on the parameters $\epsilon$ and $a$. For the
case in which the black hole is more prolate than a Kerr black hole, the photon
sphere exists only in the regime $\epsilon\leq\epsilon_{max}$ for prograde
photon. The upper limit $\epsilon_{max}$ is a function of the rotation
parameter $a$. As $\epsilon>\epsilon_{max}$, the deflection angle of the light
ray closing very to the naked singularity is a positive finite value, which is
different from those in both the usual Kerr black hole spacetime and in the
rotating naked singularity described by Janis-Newman-Winicour metric. For the
oblate black hole and the retrograde photon, there does not exist such a
threshold value. Modelling the supermassive central object of the Galaxy as a
rotating non-Kerr compact object, we estimated the numerical values of the
coefficients and observables for gravitational lensing in the strong field
limit.
