When I was a much younger man, I worked on a formula for cutting a circle into n-dimension sets in such a way that each intersection was the same area (although the intersections have weird shapes). The trick was to always include the 'null' set, and liberal use of Phi in calculating angles, so i called it a Phi chart. However, I lost all my work in a lightning strike, and this stupid and throughly unhelpful image is all I have left. That happened 6 or 7 years ago, and every now and again I see this image and tell myself that I should really finish it off, however, i'm always too busy. Maybe if I post it here someone else will have the academic freedom to figure it out :) More recently I did another kind of thing using a node/edge layout, but again it didn't have practical use for NGS data analysis and also got canned. Boo hoo.
Venn diagrams are particular cases of Euler diagrams showing all possible combinations. For more than 3 sets, this is not going to look nice because it can't be drawn with circles. For a small number of combinations, you could try Euler diagrams, for higher numbers, I find there's little visualization benefits.
You can try playing with the R package VennDiagram.