  An operator ideal is proper if the only operators of the form $Id_X$ it
contains have finite rank. We answer a question posed by Pietsch (1979) by
proving that there is no largest proper operator ideal. Our proof is based on
an extension of the construction by Aiena-Gonz\'alez (2000), of an improjective
but essential operator on Gowers-Maurey's shift space $X_S$ (1997), through a
new analysis of the algebra of operators on powers of $X_S$. We also prove that
certain properties hold for general $\mathbb{C}$-linear operators if and only
if they hold for these operators seen as real: for example this holds for the
ideals of strictly singular, strictly cosingular, or inessential operators,
answering a question of Gonz\'alez-Herrera (2007). This gives us a frame to
extend the negative answer to the question of Pietsch to the real setting.
