This problem is very easy to solve using induction. Base step : the minimum "replicative subunit". For life, that is usually a single cell. For nano-machinery, it is somewhat larger. For the sake of penciling in numbers, suppose you need a robot with a scoop and basic mining tools, a vacuum chamber, a 3d printer able to melt metal powder, a nanomachinery production system that is itself composed of nanomachinery, a plasma furnace, a set of pipes and tubes and storage tanks for producing the feedstock the nanomachinery needs, and a power source.
All in all, you could probably fit a single subunit into the size and mass of a greyhound bus. One notable problem is that there's enough complexity here that current software could probably not keep a factory like this running forever because eventually something would break that it doesn't know how to fix.
Anyways, you set down this subunit on a planet. It goes to work. In an hour, the nanomachinery subunit has made a complete copy of itself. In somewhat more time, it has to manufacture a second copy of everything else. The nanomachinery subunit makes all the high end stuff - the sensors, the circuitry, the bearings - everything complex, while the 3d printer makes all the big parts.
Pessimistically, this takes a week. A greyhound bus is 9x45 feet, and there are 5.5e15 square feet on the earth's surface. To cover the whole planet's surface would therefore take 44 weeks.
Now you need to do something with all the enormous piles of waste material (stuff you cannot make more subunits with) and un-needed materials. So you reallocate some of the 1.3e13 robotic systems to build electromagnetic launchers to fling the material into orbit. You also need to dispose of the atmosphere at some point, since all that air causes each electromagnetic launch to lose energy as friction, and waste heat is a huge problem. (my example isn't entirely fair, I suspect that waste heat would cook everything before 44 weeks passed). So you build a huge number of stations that either compress the atmosphere or chemically bond the gasses to form solids.
With the vast resources in orbit, you build a sun-shade to stop all solar input to reduce the heat problem, and perhaps you build giant heat radiators in space and fling cold heat sinks to the planet or something. (with no atmospheric friction and superconductive launchers, this might work). You can also build giant solar arrays and beam microwave power down to the planet to supply the equipment so that each subunit no longer needs a nuclear reactor.
Once the earth's crust is gone, what do you do about the rest of the planet's mass? Knock molten globules into orbit by bombarding the planet with high energy projectiles? Build some kind of heat resistant containers that you launch into space full of lava? I don't know. But at this point you have converted the entire earth's crust into machines or waste piles to work with.
This is also yet another reason that AI is part of the puzzle. Even if failures were rare, there probably are not enough humans available to keep 1e13 robotic systems functioning, if each system occasionally needed a remote worker to log in and repair some fault. There's also the engineering part of the challenge : these later steps require very complex systems to be designed and operated. If you have human grade AI, and the hardware to run a single human grade entity is just a few kilograms of nano-circuitry (like the actual hardware in your skull), you can create more intelligence to run the system as fast as you replicate everything else.
Molecular nanotechnology, or MNT for those of you who love acronyms, seems to be a fairly common trope on LW and related literature. It's not really clear to me why. In many of the examples of "How could AI's help us" or "How could AI's rise to power" phrases like "cracks protein folding" or "making a block of diamond is just as easy as making a block of coal" are thrown about in ways that make me very very uncomfortable. Maybe it's all true, maybe I'm just late to the transhumanist party and the obviousness of this information was with my invitation that got lost in the mail, but seeing all the physics swept under the rug like that sets off every crackpot alarm I have.
I must post the disclaimer that I have done a little bit of materials science, so maybe I'm just annoyed that you're making me obsolete, but I don't see why this particular possible future gets so much attention. Let us assume that a smarter than human AI will be very difficult to control and represents a large positive or negative utility for the entirety of the human race. Even given that assumption, it's still not clear to me that MNT is a likely element of the future. It isn't clear to me than MNT is physically practical. I don't doubt that it can be done. I don't doubt that very clever metastable arrangements of atoms with novel properties can be dreamed up. Indeed, that's my day job, but I have a hard time believing the only reason you can't make a nanoassembler capable of arbitrary manipulations out of a handful of bottles you ordered from Sigma-Aldrich is because we're just not smart enough. Manipulating individuals atoms means climbing huge binding energy curves, it's an enormously steep, enormously complicated energy landscape, and the Schrodinger Equation scales very very poorly as you add additional particles and degrees of freedom. Building molecular nanotechnology seems to me to be roughly equivalent to being able to make arbitrary lego structures by shaking a large bin of lego in a particular way while blindfolded. Maybe a super human intelligence is capable of doing so, but it's not at all clear to me that it's even possible.
I assume the reason than MNT is added to a discussion on AI is because we're trying to make the future sound more plausible via adding burdensome details. I understand that AI and MNT is less probable than AI or MNT alone, but that both is supposed to sound more plausible. This is precisely where I have difficulty. I would estimate the probability of molecular nanotechnology (in the form of programmable replicators, grey goo, and the like) as lower than the probability of human or super human level AI. I can think of all sorts of objection to the former, but very few objections to the latter. Including MNT as a consequence of AI, especially including it without addressing any of the fundamental difficulties of MNT, I would argue harms the credibility of AI researchers. It makes me nervous about sharing FAI literature with people I work with, and it continues to bother me.
I am particularly bothered by this because it seems irrelevant to FAI. I'm fully convinced that a smarter than human AI could take control of the Earth via less magical means, using time tested methods such as manipulating humans, rigging elections, making friends, killing its enemies, and generally only being a marginally more clever and motivated than a typical human leader. A smarter than human AI could out-manipulate human institutions and out-plan human opponents with the sort of ruthless efficiency that modern computers beat humans in chess. I don't think convincing people that smarter than human AI's have enormous potential for good and evil is particularly difficult, once you can get them to concede that smarter than human AIs are possible. I do think that waving your hands and saying super-intelligence at things that may be physically impossible makes the whole endeavor seem less serious. If I had read the chain of reasoning smart computer->nanobots before I had built up a store of good-will from reading the Sequences, I would have almost immediately dismissed the whole FAI movement a bunch of soft science fiction, and it would have been very difficult to get me to take a second look.
Put in LW parlance, suggesting things not known to be possible by modern physics without detailed explanations puts you in the reference class "people on the internet who have their own ideas about physics". It didn't help, in my particular case, that one of my first interactions on LW was in fact with someone who appears to have their own view about a continuous version of quantum mechanics.
And maybe it's just me. Maybe this did not bother anyone else, and it's an incredible shortcut for getting people to realize just how different a future a greater than human intelligence makes possible and there is no better example. It does alarm me though, because I think that physicists and the kind of people who notice and get uncomfortable when you start invoking magic in your explanations may be the kind of people FAI is trying to attract.