We often learn that the order parameter is a perfect tool for the study of a phase transition (assuming Landau-Ginzburg theory is applicable). The order parameter is finite in the ordered phase, and perfectly zero in the disordered phase. Indeed, this is how we define the order parameter; if a physical observable is finite in the disordered phase, then it wouldn't be an order parameter by definition.
However, experimentally speaking, such a perfect order parameter seems unlikely. Are there any experimental, real-world examples (in condensed matter, high-energy, statistical physics, etc.) where the order parameter is nearly zero but still finite in the disordered phase, just above the critical point?