There are plenty of quantities that do not obey the superposition principle. A simple pendulum, for example, will behave differently (with a longer period) if you double the initial amplitude.
What Griffiths means by that quote is that for the electromagnetic field there are no situations where the fields fail to add linearly. More specifically, the superposition principle is encoded in the linearity of Maxwell's equations, which states that
If $(\mathbf{E}_1(\mathbf{r},t),\mathbf{B}_1(\mathbf{r},t))$ and $(\mathbf{E}_2(\mathbf{r},t),\mathbf{B}_2(\mathbf{r},t))$ are solutions of Maxwell's equations, then $$(\mathbf{E}_1(\mathbf{r},t)+\mathbf{E}_2(\mathbf{r},t),\mathbf{B}_1(\mathbf{r},t)+\mathbf{B}_2(\mathbf{r},t))$$ is also a solution.
This is indeed consistent with experiment, except for two situations:
If the field strength inside a medium exceeds that of its linear response, then the material ("macroscopic") Maxwell equations are no longer a linear problem. This is the bread and butter of nonlinear optics, which describes a broad range of phenomena. However, this is not a failure of Griffith's claim, as the 'microscopic' fields $\mathbf{E}$ and $\mathbf{B}$ are still a linear superpositions of those created by the free and bound charges.
In certain, very careful experiments, it is possible to observe the scattering of light by light. This is explained by Quantum Electrodynamics as the temporary creation and annihilation of virtual particle-antiparticle pairs where the light beams meet, which transfer energy and information from one beam into the other. This does violate the superposition principle as stated above and as meant by Griffiths in his textbook, and it has been observed experimentally. However, outside of very specific experiments specially designed to observe it, this effect is negligible and can be ignored as regards classical electrodynamics. In the quantum version, you have a whole host of such problems to deal with.