The $q$-character is a strong tool to study finite-dimensional representations of quantum affine algebras. However, the explicit formula of the $q$-character of a given representation has not been known so far. Frenkel and Mukhin proposed the iterative algorithm which generates the $q$-character of a given irreducible representation starting from its highest weight monomial. The algorithm is known to work for various classes of representations. In this note, however, we give an example in which the algorithm fails to generate the $q$-character.