Jumper cables, road flares, electric tire air pump, first aid kit, tire repair kit, pressure can of tire goo. All the usual break down stuff...but then I don't carry a spare. Tire on the roof basket will really eat into the gas mileage...but then so will the basket empty.
Breakdown kit is a good idea, even just to help others. In my old CRV the spare is mounted on the rear door. Rather than cover up the spare wheel well, Honda converted it to a "wet well" with a folding picnic table on top so if you go surfing, take your kids to the beach/lake/local pool then there's somewhere to put all the wet/sandy gear or just use it for extra storage when camping/travelling. Lining it with a waterproof liner seems like a good multi-purpose option.
You get a very brief glimpse at about 1:36 in this vid
I kept the spare in place but built a raised deck to have room for my jumper cables and other stuff. Not hinged - just sitting on wood feet in strategic locations. An inexpensive small carpet from WallMart sits on top. Never liked the lack of space between the spare and the factory deck, and when I load/unload gear, I never liked the inset of the factory deck anyway.
I dont carry one either. For daily driving, I can call AAA/roadside assistance. If im going off-road/exploring, I have a full sized matching 5th wheel/tire that I put on a carrier I built for the hitch.