Let A be a selfinjective connected Nakayama algebra. What is the Loewy length of any indecomposable projective Amodule?

The Loewy length can be arbitrary. See e.g. [Assem, Simson, Skowronski: Elements of the Representation Theory of Associative Algebras 1] Proposition V.3.8:
The Loewy length of each projective indecomposable should then be $h$. This certainly also holds true for nonbasic algebras if you replace isomorphism by Morita equivalence. 

