This totally ignores the possibility of using different airplane designs altogether. Build something whose optimal speed is slower, and you can make the whole thing more efficient.
Probably the best single-number measure of aerodynamic efficiency is the lift-to-drag ratio. For most airliners, 787 most likely included, this ratio is in the low 20s. The best machines out there are in the 70s. Of course, they fly slow, which is a tradeoff airliners don't want to make, but it shows that there's plenty of room for improvement.
Can we build a more efficient airplane? Of course! But we may not want to.
See http://en.wikipedia.org/wiki/File:Drag.jpg for the different types of drag on an aircraft (the 'pushing stuff through the air' drag is called parasitic, and the 'side-effect of lift generation' drag is induced).
The problem is that by doing this, you increase the structural mass of the wing for a given lift as you need a longer wing. There is a point on the optimisation curve between increase in efficiency from AR & the increase in structural weight from longer wings. Boeing and Airbus have teams of people who do this stuff for a living.
There are also human factors involved, like airports being set up for a maximum wingspan, and people being unlikely to choose an airline with slower flights.
"To estimate the energy required to move freight by plane, per unit weight of freight, ...then [a 747's] transport cost is 0.45 g, or roughly 1.2 kWh/ton-km. This is just a little bigger than the transport cost of a truck, which is 1 kWh/ton-km....
"...This is a bit more efficient than a typical single-occupant car (12 km per litre). So travelling by plane is more energy-efficient than car if there are only one or two people in the car; and cars are more efficient if there are three or more passengers in the vehicle."
The original article is pretty bad; the answer to the title's question, "Can we build a more efficient airplane?" is nothing but a link to MacKay. The rest of the article entertaining lead-up fluff.
For example, an airplane with a 40:1 L/D (done in the 70s, easy today) able to carry half its empty weight in payload (nothing too hard there) will require about 0.3kWh/mile per ton of payload to maintain level flight. Even accounting for propulsion inefficiencies, that's way more efficient than the 747.
Why don't we have such airplanes? Not because it's somehow physically impossible, nor even technologically impossible, but simply because it's not worth the tradeoffs. Nobody cares about increasing the efficiency of air travel if the result is an airplane that's ten times slower than a modern airliner.
Can we build a more efficient airliner? Of course! To say "no" is to imply that modern airliners are optimized for efficiency above all other things, which is pretty much ridiculous on its face. Can we build a more efficient airliner while still keeping all the other stuff we want out of airliners, like a mach 0.8 cruising speed and global range? Well, that's a bit harder.
The whole reason we use jet aircraft is because it allows us to travel faster for the same amount of fuel. Passenger transport of all forms is mainly about pushing air out of the way and the air at 40,000 feet is very thin. The fuel you use staying airborne is more than paid back by the fuel you save in aerodynamic drag due to the thinner medium.
If speed is no longer a priority, then that tradeoff collapses. Keeping people comfortable for long periods requires mass, which is very expensive to put in the air, but practically free to move by sea or rail. You're then left offering passengers the choice between two equally efficient options - spend three days in an economy class seat, or three days in a sleeper cabin with a full-sized bed, a shower, room to stretch your legs and a view from the window.
I think conventional airplanes have reached their limit. That does not mean airplanes cannot become more efficient. BWBs (Blended Wing Body) are more efficient than traditional aircraft. They have increased payload and fuel efficiency. Kind of like a merging of a flying wing (B2 Bomber) and traditional swept wing (747/777).
"Blended wing body has lift-to-drag ratio 50% greater than conventional airplane."
Boeing/NASA are currently exploring BWB with the X-48.
The efficiency improvements come from all over the place including dealing with wing tips (fences, winglets, "sharklets"), weight reductions (lighter materials, redesigned components) and engines (weight, gearboxes, compression ratios).
The plane manufacturers and engine manufacturers are constantly doing tweaks to give improvements, usually managing around 1% per year. Search for "performance improvement package" to see numerous press releases and articles.
In the future there are blended wing bodies and open rotor engines that have another leap in efficiency.
 The tradeoffs are very similar to what we see in the IT industry. For example the 737 is lighter and sits lower to the ground. That means it can't use larger diameter more efficient engines. And makes it more efficient for shorter routes due to less weight, but the heavier higher A320 can then be more efficient over longer routes. Every change made involves retooling and recertification so it isn't a simple decision to just do everything possible. And newer better planes effect the residual values of older planes which makes lessors potentially less likely to buy your newer planes due to economic uncertainty. There was a lot of debate and speculation about re-engining the existing models, re-winging too, fuselage material changes etc.
That's all fine in the current model, but other modes of transport are thinkable. For example an airship that doesn't need to spend energy on the lift, or a ballistic transport model (think ICBM) that uses flight parabolas around the curvature of the earth.
Ok, you probably wouldn't call their "airplanes" anymore, at least not by traditional terminology.
Update: oh, and you can always try to improve the constant factors involved, like the total aerodynamic drag and weight
If you are going for crazy, why not use a nuclear reactor? http://en.wikipedia.org/wiki/Convair_X-6
Common sense says you should compare two flights covering the same distance. The flight that's twice as fast takes half as long, so you end up with the energy required being proportional to the square of velocity.
(The power required is still proportional to the cube of speed, which is why e.g. a Bugatti Veyron costs so much.)
Weight reduction - Reduce the weight and you can reduce the air you need to move.
There are also conceptual designs for integrated wing aircraft that if they were easier to make and could be made to work with current airports might have further real gains.
These added to the engines and other factors mentioned probably make at least a 40-50% improvement possible (although not necessarily easy or quick to achieve).
This doesn't break the concepts explained or mean that there is unlimited room for improvement but 10% is not going to be the limit by a long way.
A modern sailplane can fly at higher speeds than those with thicker wing profiles with less drag, optimizing the lift. This is an approach that modern airliners could follow, optimizing the lift and reduce the drag, to save on fuel, not reducing the speed.
The engine is not the only thing that can be optimized.
The optimal height of an aircraft, however, depends on the mass of itself and the payload.
(I'll have to defer commenting to someone with knowledge of physics than me)
It's also a great book
also, the SR-71 is very efficient considering it can cruise at mach 3.2, which is a pretty insane feat.
Quote: What you find is that it really just depends on a few factors: the shape and surface of the plane, and the efficiency of its engine.
It would also depend on weight. New planes are also lighter, as article mentions (eliminated aluminum sheets and fasteners).
Some other weight could be eliminated by replacing in-flight entertainment screens in every chair and wires required for them with tablets + wireless.
High-speed rail and tubes are the future of transport.
This makes for a funny* situation regarding things like CA high speed rail plan for SF to LA which is currently slated for completion in 2028. If this turns out to be 10 years after the first commercial robotic car, it will likely be obsolete prior to its first run.
*Depends on your sense of humor.
I have no idea what you are referring to by tubes though.
Probably a reference to Elon Musk's recent claim that he had come up with some secret technology that could dramatically improve long distance transportation, with much (much) higher speed, lower energy usage, and (much) lower cost than conventional HSR.
Unfortunately the only detail Musk would give was a very hand-wavy reference to "tubes," and a subsequent clarification that he didn't mean the usual sort of "evacuated tube transport" people have talked about for ages and ages.
Based on all the discussion I've seen, though, it seems pretty likely it's just a vague idea Musk had that he didn't bother to think through, and it will fall apart when he tries to fill in the details...
Using HSR to facilitate movement within the high-density regions can make sense, even if using it to facilitate movement from one high-density region to another doesn't.