What's the biggest building we could build on the world?
When you run out of planetary mass. The entire world can be turned into a great disc-shaped building.
1) Build an upside-down track around the world. On that track you build a maglev train that wraps all the way around. Evacuate the space it is in, accelerate the train. Once it's moving above orbital velocity it will exert an upward force on the track which can support it. While you are building this you will need some sort of temporary supports in the ocean or if you are going to be strict enough about the can't-touch-water then onto artificial islands. Once the train is spinning they could be removed.
2) Once you have this basic structure you can expand upwards at will and within reason you can expand sideways. (A sideways expansion will exert some lateral force and thus you're not free to expand infinitely this way.)
3) As you convert more and more of the planet to the building you can also build down into the space where you removed the mass.
Huge.
Imagine a building that covers every cm of land in the world. Imagine this building extending kilometers into the sky. Imagine it extending kilometers into the ground. Imagine draining bodies of water, freezing them or simply building over the top of them.
If one has limitless resources, then one can do impressive things.
148,326,000 km2 of surface available on Earth can be covered in the scenario you're talking about. Materials would have to be diverse (wood for when we don't have steel, etc.), and I imagine on your roof you would have agriculture, wetlands, forestry, utilities, and air strips. A percentage of your land would be removed because they're too steep of incline to cover, and another percentage for your major utilities such as nuclear plants. The latter can be built on artificial islands, perhaps.
You could build something like an ant colony.
Start by having the nano machines hollow out a bunch of tunnels and chambers. With all the material removed for this you could add another layer of tunnels and chambers on the surface. If you dig deep enough you could start using some of the mantle as material as well. and keep building layers. In this way the whole world could be a giant structure.
This way you still have the world remain intact, so you don't have to worry about converting it into a giant ring in space. The tunnels and chambers are all circles and domes so they support themselves extremely well and you could build many many layers of tunnels and chambers. Imagine the world covered in a set of tunnels and chambers up as high as as our atmosphere can support us.
This is kind of a mega structure. It's feasible with nano machines doing all the work and doesn't require special technology to build.
Assuming we are not employing "ultra tech" and handwavium, the size of any megastructure on Earth will be limited by the structural strength of the materials. For practical purposes, we will probably use things which are relatively common and inexpensive, so your megabuilding will be built using structural members made of steel.
I have seen various designs for more or less realistic megastructures made of steel and concrete, Paolo Soleri had conceptual designs for arcologies built this way in his book "Arcology: City in the Image of Man" in 1969, including megastructures like the Hexahedron and Arcube which were to be over a kilometre high and have a similar width and depth
With modern techniques, it seems possible to build even larger structures, steel towers could be many kilometres tall, serving as the basis of launching tracks for space vehicles, for example.
Without a clear definition as to the reason and purpose of the building, I will say that a 5×5 kilometre cube does not seem beyond the bounds of possibility for steel construction of a building; a sort of mega arcology structure.
