  We study the Kodaira dimension of the compactified n-fold Kuga variety over
the moduli space of principally polarised abelian g-folds. We construct a
suitable compactification, which we call a Namikawa compactification, and show
that in most cases it has canonical singularities. We then use results about
the slope of Siegel modular forms to determine the Kodaira dimension for all
g>1 and n>0.
