While higher-order approximations exist and are crucial to a better understanding and description of reality, they are not typically referred to by number.
Continuing the above, a third-order approximation would be required to perfectly fit four data points, and so on. See polynomial interpolation.
These terms are also used colloquially by scientists and engineers to describe phenomena that can be neglected as not significant (e.g., "Of course the rotation of the earth affects our experiment, but it's such a high-order effect that we wouldn't be able to measure it" or "At these velocities, relativity is a fourth-order effect that we only worry about at the annual calibration.") In this usage, the ordinality of the approximation is not exact, but is used to emphasize its insignificance; the higher the number used, the less important the effect.