What does non degeneracy mean for a preference? I saw a non-degeneracy assumption in Gilboa and Schmeidler's paper (maxmin expected utility with non-unique prior).

The statement is "Not for all $f$ and $g$ in $L$, $f \geq g$".

So can you explain the intuition for this assumption and give some example so that I can understand the "non-degeneracy" property better?
It means what Herr K. has guessed: there are acts $f$ and $g$ that the decision maker does not deem equivalent from the preference standpoint. In other words, the DM expresses a strong preference in at least one comparison he makes. a preference that does not satisfy it is a AA subjective expected utility with a constant index $u$ (i.e. such that the representation is $\int u dp$ for an arbitrary belief $p$). On the other hand if at least for one comparison the DM says $f$ is strictly better than $g$ then you satisfy it.