The forest fire model in statistical physics represents a paradigm for systems close to but not completely at criticality. For large tree growth probabilities p we identify periodic attractors, where the tree density ρ oscillates between discrete values. For lower p this self-organized multistability persists with incrementing numbers of states. Even at low p the system remains quasiperiodic with a frequency ≈p on the way to chaos. In addition, the power-spectrum shows 1/f^{2} scaling (Brownian noise) at the low frequencies f, which turns into white noise for very long simulation times.