If by "double jointed", you mean hyper-mobility, or hyper-flexion, which is characterized by abnormal joints which flex further than normal, then no. I've done some research and could find no martial art which requires such skill or curse.
As to whether there is a style where you might perform better by being hyper-mobile, then no, I could find no such style here, either, despite that some schools can encourage it.
As a note, I would say most styles - be it martial, aesthetic, or spiritual - do not require hyper-mobility, but some strive for ultra flexibility.
For example, some styles of yoga have hyper-mobility as a goal. Nonetheless, such capability is not required, and usually, is not beneficial. In combat styles like Taekwondo, such flexibility is occasionally desired, but it is not a requisite. In fact it can be detrimental. Being hyper-mobile doesn't necessarily mean "strong", and often, one is very weak (and very compromised) at these extreme ranges of motion. Were you to shore up that weakness, you lose flexibility.
So it is a trade-off: what are your goals? If for aesthetics, then by all means, go for hyper-mobility. Go to extreme yoga classes, buy Chuck Norris' stretch machine, or straddle across two chairs.
But if you are looking for sport competition or self-defense, then such mobility can be harmful to your goals.