We investigate experimentally the nonlinear structures that develop from interacting vortex rings induced by a sinusoidally oscillating ellipsoidal disk in fluid at rest. We vary the scaled amplitude or Keulegan-Carpenter number $0.3$<$N$=<$1.5$, where $A$ is the oscillation amplitude and $c$ is the major diameter of the disk, and the scaled frequency or Stokes number $100$<$\beta =fc$$\mathrm{<mrow\; data-mjx-texclass="ORD">2</mrow>}$$/\nu$<$1200$, where $f$ is the frequency of oscillation and $v$ is the kinematic viscosity. Broadly consistent with global linear stability analyses, highly organized nonlinear structures with clear azimuthal wave number emerge as sequential vortex rings are shed from the disk. These organized structures exhibit wave numbers ranging from $m$=$2$ to $m$=$9$ and can be further divided into two distinct classes, distinguished by the phase and symmetry properties above and below the disk. We find some discrepancies between experiments and linear stability analysis, due to the inherent nonlinear mechanisms in the experiments, particulary on the boundary between the two branches, presenting unevenly distributed flow structures along the azimuthal direction.