  For any smooth irreducible projective curve $X$, the gonality sequence $\{d_r
\;| \; r \in \mathbb N \}$ is a strictly increasing sequence of positive
integer invariants of $X$. In most known cases $d_{r+1}$ is not much bigger
than $d_r$. In our terminology this means the numbers $d_r$ satisfy the slope
inequality. It is the aim of this paper to study cases when this is not true.
We give examples for this of extremal curves in $\PP^r$, for curves on a
general K3-surface in $\PP^r$ and for complete intersections in $\PP^3$.
