For strict-feedback nonlinear systems, this note shows that it is impossible to reduce to zero the optimal cost in the regulation of more states than the number of control inputs in the system, even using unrestricted control effort. By constructing a near-optimal cheap control law, we characterize the infimum value of the optimal regulation cost as the optimal value of a reduced-order regulator problem. We illustrate our results with an example on the optimal control of a magnetic suspension system.