  We study a generic model of a Chern insulator supplemented by a Hubbard
interaction in arbitrary even dimension $D$ and demonstrate that the model
remains well-defined and nontrivial in the $D \to \infty$ limit. Dynamical
mean-field theory is applicable and predicts a phase diagram with a continuum
of topologically different phases separating a correlated Mott insulator from
the trivial band insulator. We discuss various features, such as the elusive
distinction between insulating and semi-metal states, which are unconventional
already in the non-interacting case. Topological phases are characterized by a
non-quantized Chern density replacing the Chern number as $D\to \infty$.
