How are J1407b's rings possible? If planetary rings are maintained by staying within the Roche limit, how is it that  J1407b's rings extend so far? Surely the gravity and tidal forces should be low enough to allow a moon to form.
There are a few ways that a gigantic ring system outside the Roche limit might be possible.   Ring systems can extend well past the Roche limit if they are very faint or very young.   A faint/light ring system doesn't have enough mass to coalesce and a young system might not have enough time.   Young small moons with lots internal heat of formation can be volcanically active too, feeding a ring system.  

Saturn's E ring is well outside it's Roche limit.

https://upload.wikimedia.org/wikipedia/commons/f/f7/Saturn%27s_Rings_PIA03550.jpg

The most likely explanation for it's enormous and very dense rings, however is that J1407b is very young and a lot of it's ring system will likely to coalesce into moons in time.  The breaks in it's ring system suggests that some moons are already clearing paths in it's rings.  

It's mass is also very significant.   While difficult to estimate, it's thought to be about 10-40 Jupiters in mass.  Source.  That would give it a mass in the range of 33 to 130 Saturns and a Roche limit some 3 to 5 times (cube root of the mass), distance to Saturn's center, perhaps 6-10 times the distance from the surface surface.   Because it's ring system is about 200 times the size of Saturn's rings, the mass alone wouldn't explains part of it.    Mostly likely it's a combination of very massive planet/perhaps a brown dwarf star and young plant/star with a satellite system still in formation.  

There might also be a very powerful magnetic field at play that helps prevent the ring debris from forming into satellites, but that's just speculation on my part.