We have searched for anomalous Z --> gamma gamma gamma events with the L3 detector at LEP. No significant deviations from the expected QED e(+)e(-)-->gamma gamma gamma events are observed. The branching ratio upper limit for a composite Z decaying directly into three photons is found to be 1.0 x 10(-1) at 95% C.L. The branching ratio upper limits for the process Z-->gamma X,X-->gamma gamma are in the range of 0.4 to 1.3 x 10(-5), depending on the mass and width of the scalar particle X. In the context of a model with magnetic monopoles coupling to the Z, we find BR(Z-->gamma gamma gamma) < 0.8 x 10(-5) at 95% C.L; this results in a lower mass limit of 510 GeV for a magnetic monopole.