  The k-planar graphs, which are (usually with small values of k such as 1, 2,
3) subject to recent intense research, admit a drawing in which edges are
allowed to cross, but each one edge is allowed to carry at most k crossings. In
recently introduced [Binucci et al., GD 2023] min-k-planar drawings of graphs,
edges may possibly carry more than k crossings, but in any two crossing edges,
at least one of the two must have at most k crossings. In both concepts, one
may consider general drawings or a popular restricted concept of drawings
called simple. In a simple drawing, every two edges are allowed to cross at
most once, and any two edges which share a vertex are forbidden to cross.
While, regarding the former concept, it is for k<=3 known (but perhaps not
widely known) that every general k-planar graph admits a simple k-planar
drawing and this ceases to be true for any k>=4, the difference between general
and simple drawings in the latter concept is more striking. We prove that there
exist graphs with a min-2-planar drawing, or with a min-3-planar drawing
avoiding crossings of adjacent edges, which have no simple min-k-planar
drawings for arbitrarily large fixed k.
