Not counting the obvious (car with gas etc.), the bare minimum would be a decent map and a cell phone. The cell phone is mainly for safety in case of breakdown or other emergency and for reporting severe weather. A cell phone could also be used to receive nowcasting support or in more advanced models, used to access data. A good map is also a necessity. I chased for many years without GPS and I still use maps as back up and plotting routes. One might add a weather radio for warnings though local stations will often carry weather warnings. I chased with maps, cell phone, and weather radio for a few years before getting a laptop.
Laptops, GPS devices, temp and dewpoint monitoring devices, XM are all nice and may improve success but are not necessary. I would even say that a camera or camcorder is not necessary. I have a friend that has travelled to many countries and has never taken a picture or shot a video as he prefers to enjoy the trips as is and rely on memories. Not my style but could be applied to chasing.
Bill Hark