  The graph crossing number problem, cr(G)<=k, asks for a drawing of a graph G
in the plane with at most k edge crossings. Although this problem is in general
notoriously difficult, it is fixed- parameter tractable for the parameter k
[Grohe]. This suggests a closely related question of whether this problem has a
polynomial kernel, meaning whether every instance of cr(G)<=k can be in
polynomial time reduced to an equivalent instance of size polynomial in k (and
independent of |G|). We answer this question in the negative. Along the proof
we show that the tile crossing number problem of twisted planar tiles is
NP-hard, which has been an open problem for some time, too, and then employ the
complexity technique of cross-composition. Our result holds already for the
special case of graphs obtained from planar graphs by adding one edge.
