Thanks, @encephalon. That’s a good question. Before I go into my Ted Talk and share some documentation I found…
Disclaimer:
This will obviously be a wild conceptual approximation. I of course don’t have the resources to run it through the unbelievably expensive CFD analysis cycle it would need to design it properly (which usually takes a warehouse-sized server to compute). Also, I’m not an aerodynamicist by trade or a designer for Boeing. So the exact reasons and design data would be speculation and if I was a Boeing employee, I would be compelled by a serious NDA to never talk about heavily guarded trade secrets.
So I’m purely approaching this like any good professional designer would do…act like I know what I’m talking about and give it my best guess based on the best possible research and intuition 
That being said…let’s have some fun:
You asked, "anything you can tell us about the tip of the wings?"
The “short” answer is that the raked wing tip is designed for better takeoff performance and fuel efficiency of the wing at a very wide range of situations. This is an alternative to winglets found on contemporary commercial airliners such as the Airbus A350 (picture below)
The long answer is: Don’t think of the raked wing tip as its own thing. Think about the design of the entire wing (aerofoil) as a whole for it to make sense.
The advanced composite 787 wing is what we call a “supercritical aerofoil”. This basically means the shape of the wing appears to have a “flat top” compared to more traditional top-rounded wing cross sections.
The wing shape (swept at 32.2 degrees) is designed precisely for what is known as a “long operational range”, which means it needs to perform a wide amount of tasks at various conditions an airliner will encounter in its life such as varying lengths of travel, varying speeds, varying altitudes, turbulence, various use cases, various configurations, economic conditions, extreme ambient conditions, variable weight loading, very dynamic performance requirements at various airports, etc etc.
The wing tips sweep back even further to complete the design of the 787 wings, versus a winglet which was traditionally an option (although not so much anymore) that was attached and integrated into the wing after the fact to give the plane better performance.
Basically, this wing design is part of what will make the 787 live a very potentially long life (easier to justify for longer) in an uncertain future in aviation compared to previous generations of aircraft.
Because the 787 wings flex so much depending on load, the wing tips also have to perform well in various camber (flex) configurations of the entire wing. This means the design has to perform well in flight when the wing is minimally loaded (minimal flex) all the way up to a fully loaded wing (maximum flex of the wing). So on the 787, the tip shape is exactly the shape it needs to be to work under all these possible conditions. So as for the precise shape of the real 787 wing tip, unless anyone here:
a) has an ultra-high resolution scanner and a 787 laying around
or
b) is a design engineer for Boeing and doesn’t mind destroying their career and Boeing along with it talking about the true details of the wing design
It’s all best guess
Some light reading that might help:
https://www.boeing.com/commercial/787/by-design/#/smooth-wing-technology
