  We study the dynamics of smooth interval maps with non-flat critical points.
For every such a map that is topologically exact, we establish the full
(level-2) Large Deviation Principle for empirical means. In particular, the
Large Deviation Principle holds for every non\nobreakdash-renormalizable
quadratic map. This includes the maps without physical measure found by
Hofbauer and Keller, and challenges the widely-shared view of the Large
Deviation Principle as a refinement of laws of large numbers.

Dynamical Systems;Probability