Generalizations of planar graphs that include hypercubes with large side length in $R^d$
You may want to look at nowhere dense graphs.
http://www.sciencedirect.com/science/article/pii/S0195669811000151

One of the reasons why minor-closedness is natural is the following. We typically want to work with families of graphs rather than specific graphs. And we want to solve problems with arbitrary weights/capacities on edges/nodes. Suppose we want to solve the shortest path problem in a family of graphs. Then, if we allow for zero length and infinite lengths then basically we are allowing minor operations on the family. In some settings it makes sense to work with unweighted graphs where positive results can be obtained for larger families of graphs that are not necessarily minor-closed.