The Fr\"olicher spectral sequence can be arbitrarily non degenerate

The Fr\"olicher spectral sequence of a compact complex manifold $X$ measures the difference between Dolbeault cohomology and de Rham cohomology. We construct for $n\geq 2$ nilmanifolds with left-invariant complex structure $X_n$ such that the $n$-th differential $d_n$ does not vanish. This replaces an earlier incorrect example by the second author.