A little digression on money and what it represents
First of all, money itself has little value. We value money because our society has established it as an abstract representation for labor + materials.
Thus projects requiring expensive materials or lots of labor cost more money.
So how does this affect your question?
Assume Project X might be humanity's #1 goal and we might throw all sorts of labor at it.
- We have to understand that many many ordinary activities must
continue (farming, water workers, road construction, etc.).
- Plus, we may want to transport unlimited materials for a project,
but we have limited ability to transport materials.
- The project may require special materials (e.g. Platinum as a
catalyst) and it will compete with other necessary activities that
we are still doing.
During special times, governments have severely limited "normal" activities to devote more effort towards the "special" activities (think rationing during WWII).
So you saying assume infinite money, is the equivalent of saying assume infinite labor and resources. Which limits us only to what our current state of science says is practical.
The case of the Pyramid 100 km tall
Lets look at your case, a pyramid 100 km tall. Our materials and structural knowledge are immensely greater than ancient Egypt's. But I don't need to know the materials that we'd use in such a pyramid because I can look at the Earth for equivalent natural formations and find none.
Why is this?
Because under immense pressure, the Earth's materials flow like plastic. If you built a big enough structure, you'd cause the crustal plate it had been built on to begin sinking into the Earth (the concept is called Isostacy). As a very rough approximation, I'd look at the tallest natural structures and use that as the order of magnitude guess maximum size of man-made structures.
So perhaps 10km is doable given infinite man-power and resources but it couldn't get much taller.
Other Candidate Projects
I've also thought that megastructures on Earth weren't all that interesting, so I included some other projects that I'd expend our infinite labor and resources upon:
- National Aerospace Plane
- Cheap Space Launch Infrastructure
- Fusion Power Plants
- Interplanetary Spacecraft
- Space Colonies
- Space Elevator
- Interstellar Spacecraft
- Terraform Mars
- Terraform Mercury
- Terraform Venus
1. National Aerospace Plane
In the late 1980s I worked on the National Aerospace Plane. At the start, of the project, I felt that we could not such a thing. By the end, I had come to the conclusion that we could build it, if we sunk enough resources into it.
Estimated time to complete 5-10 years.
2. Cheap Space Launch Infrastructure
R&D and construction for Light Craft / laser launcher, Ram Accelerator, Light Gas Gun, or Coil Gun to launch objects into space cheaply.
Estimated time to R&D 5-15 years.
Estimated time to construct 5-10 years.
3. Fusion Power Plants
We still don't understand everything that we need to build viable fusion power plants but after 60 years of trying we are finally at least in the ballpark. At current project funding, such a plant won't reach scientific break-even for a decade or so. Within 5-10 years after that we might achieve engineering break-even. Economic break-even will follow sometime after that.
Unlimited resourced would speed this up significantly, perhaps halving the time it takes to finish.
Estimated time to complete 10-20 years.
4. Interplanetary Craft
Using current state of the art nuclear power and propellants, we could build space craft that would scoot to Saturn within months rather than 6 or more years.
Estimated time to complete: 10-20 years
5. Space Colony
Using the craft mentioned above, there are no theoretical reasons that we could not build, populate, and maintain colonies on many bodies of the solar system.
Estimated time to complete 20-30 years.
6. Space Elevator
Until about 15 years ago, space elevators weren't even deemed scientifically feasible. Oh sure, the required materials strength was within the theoretical maximum chemical bond strength of materials we knew (e.g. single crystal iron could do it), however, we didn't know of any bulk materials with the sort of tensile strength required for the project.
Then we discovered how to make nanotubes. Single nanotubes have the desired strength and light weight properties for a project like this. However, we still don't know how to make single tubes with a length of 48,000+ km long.
But imagine that we develop a method to grow the nanotubes in-situ microgravity. The main problem facing a project after resolving that issue would be the politics :(
Estimated time to complete R&D 20+ years
Estimated time to complete construction 10-20 years
7. Interstellar Craft
Some of the most advance propulsion schemes current known (e.g. Nuclear Pulse Propulsion [aka the old BOOM - BOOM], Fission Fragment Rocket, & Fusion Fragment Propulsion) to be possible would make manned interstellar flights possible. However, it would take most of the industrial output of the entire Earth for many decades to put it together.
Estimated time to complete 50-100 years to build
Estimated trip time 150+ years
8. Terraform Mars
Why stop with basic outposts when physics tells us what we need to do to make Mars a "shirt sleeve" environment. Use our interplanetary craft and begin playing Solar System billiards.
Shove smallish ice blocks from Saturn's rings so that they impact at Mars' poles. We bombard the poles for two reasons,
- We don't want to significantly alter Mars' rotation and
- Mars already has significant quantities of volatiles locked in its
polar caps and any volatiles we liberate from the poles will reduce
the quantities we need to move around.
While we're at it, ring it with solar reflectors to warm the planet up. It is the safe movement of the blocks of ice from Saturn's rings that takes all the time but we know of no reason that this can't be done now.
Needs to be warmed, $ H_2O $, $ N_2 $, and $ O_2 $.
After taking care of those physical and chemical needs, we need to inoculate Mars with biota that will create a biosphere we can live in.
Estimated time to complete 500+ years (but more likely 1000+ years).
9. Terraform Mercury
A more difficult project but one we could complete given enough resources and time. Move Mercury out to the current orbit of Mars. It is necessary to put Mars and Mercury in a cooler region so their atmospheres don't blow off too fast. Also, set up Mercury and Mars to orbit each other as the solar system's first true double planet (and it'll provide an amazing view!).
Moving Mars is done indirectly by means of a "gravity tractor". It takes a very long time to set up this infrastructure. Start by putting engines on a small sized asteroid (probably more than one) in one of Jupiter's Trojan points. Shove that asteroid on a course that slingshots it past Jupiter onto a course that takes it by a mid to large sized main Asteroid Belt asteroid (say Vesta). Use trajectories to alter Vesta's orbit so that it intersects Jupiter's. Ultimately, we want Vesta to perform repeated gravity assist flybys between Mercury and Jupiter. Mercury flybys will increase Mercury's momentum by robbing it from Vesta. Jupiter flybys will increase Vesta's momentum by robbing it from Jupiter. We'll use our small Trojan Asteroids as sheepdogs to make course corrections to Vesta.
Begin bombard it with ice from Saturn's rings as with Mars after cooling the surface. If you are skillful, the bombardment can also serve to spin the planet faster. Increasing Mercury's rotation (for the correct day/night cycle) may also significantly strengthen Mercury's magnetic field. Mercury already has a "strong" magnetic field (about 1% of Earth's) so its atmosphere will be somewhat protected from the solar wind.
Since at the beginning of the effort Mercury is too hot, and bombarding it with ice will add energy, it might be smart to provide it with a Sunshade to cool it off. If you are done with the Solar reflectors at Mars, just move them over and reuse them as umbrellas. If not, move them anyway but reflect light away from Mercury and towards Mars.
Needs to be moved, cooled, spun-up to a 24 hours day, $ H_2O $, $ N_2 $, and $ O_2 $.
After taking care of those physical and chemical needs, we need to inoculate Mercury with biota that will create a biosphere we can live in. Also by this time Mars' terraforming should be significantly ahead of Mercury's. It might provide a convenient base of operations for the Mercury effort.
This might allow us, in time, to have 3 inhabitable planets (Earth + Mars + Mercury).
Estimated time to complete 750+ years. I used approximately the same time to move as Mars, but I added some time to adjust the planet's rotation and volatiles - more likely 1250+ years).
10. Terraform Venus
A much more difficult project but one we could complete given enough resources and time. Since at the beginning of the effort Venus is way too hot and bombarding it with ice will add even more energy (and we won't be moving it away from the Sun), we must provide it with a Sunshade to cool it off. Assume you are done with the Solar reflectors for Mars (or Mercury) and just move them over and reuse them as umbrellas.
If we cool Venus low enough, it's $ CO_2 $ will freeze into convenient to handle blocks of dry ice. This makes ridding Venus of its excess $CO_2 $ "easy", just cut it up, load it onto a rocket, and launch it into space on something like the giant sized Orion which can launch 8,000,000 tons per launch. You could send some of it to Mercury and Mars to supplement those planets volatile inventory. I'm not sure what to do with the rest.
Bombard with water ice from Saturn's rings as with Mars. But like the terraforming of Mercury, use that bombardment to spin the planet up to a decent day / night cycle. I guess this means you bombard the equator with ice while you mine the poles for $ CO_2 $ and launch from there.
Needs to be cooled, spun-up to a 24 hours day, $ H_2O $, $ O_2 $, and $ CO_2 $ removal.
After taking care of those physical and chemical needs, we need to inoculate Venus with biota that will create a biosphere we can live in.
This might allow us, in time, to have 4 inhabitable planets (Earth + Mars + Mercury + Venus).
Estimated time to complete 1000+ years (approximately the same time to move as Mars, but longer to adjust the planet's climate - more likely 1500+ years).
What to do with Vesta when you're done? If you plan to do more Solar System billiards (e.g. Moving Io or Europa), then park it at one of the Jupiter Trojan points. If you're done with the Solar System billiards, then put it at a convenient point for Solar System infrastructure (that sucker is probably loaded with useful metals).
NOTE: I ordered these project based upon the time to complete the project. It will take longer to terraform planets than to build an interstellar spacecraft. However, I'm pretty sure the construction of an interstellar spacecraft would require more labor and resources - as in the entire world's industrial output for 50 or more years.