What properties do you need for building a tower? When I was a boy I used to daydream about building a tower so tall that the top of it would project into near space.
There would perhaps be a zero gravity area in the penthouse where my friends and I could bounce around and play space versions of various earth-based games and sports in most excellent zero-g conditions.
Much to my continued disappointment and despite all the technological advances of the last thirty or so years, no one has built such a structure.
Can anyone explain the physical limitations/constraints that are preventing someone from realising my fantasy of a 'Space Tower'?
UPDATE: This Kickstarter Project seems to be pretty confident ...
 A: First off, the limitation is a material that would not collapse under the weight - earth crust is not quite hard enough. Buckling and other instabilities, nope. Generally, forget a tower built on earth. Not a chance, no such material.
Start building from geostationary orbit and extend the "rope" both inside and outside the orbit. The outside may be just heavy counterweights, as the inside will begin to pull towards earth. Make the orbital part thicker to support extra weight, as you extend the lower part, until it reaches earth surface below.
Now the problem is the material. The only material in existence with sufficient weight-strength ratio is buckytubes. These are currently centimeters long at most, extremely expensive and you'd not only need thousands of kilometers of them... the rope to sustain its own weight would have to be about 1km thick in the thickest place (near the geostationary orbit).
Now consider:


*

*earth carbon supply, I don't think all coal mines combined could mine that much carbon

*construction craft fuel. This all would have to be lifted high enough. A LEO rocket takes many times more fuel than its payload weight. A geostationary orbit rocket - much more. The good news is the fuel can be hydrogen+oxygen which is water, and we have that aplenty. The bad news is you need at least as much energy to separate them as you gain from burning them, so the power consumption for fuel production would exceed whole world's power production.

*environmental impact of that much steam released into atmosphere

*account for micrometeorites that can really rain over your parade. And this thing being that big, collisions WILL happen. Also account for space junk.

*account for winds and storms once you reach the atmosphere. Also, upper atmosphere is pretty hot... not nice work environment, also nanotubes aren't extremely fire-proof.

*cost and impact on economy. Coal becomes super-expensive and we look for alternate sources of carbon.


And when you finally build it, calculate how long a lift travelling some 300km/h would take to reach 37,000km of the 0-gravity orbit...
EDIT:
I can't currently find the article that listed 1km thickness, but let us try to calculate parameters of the tower merely strong enough to sustain itself.
The nanotube tensile strength is $UTS=6422kg/mm^2$ (1)
The density is $\rho=1.4g/cm^3$ 
The ribbon is said to be 1m wide.
The thickness will vary. For the needed $M_0=20t=20000kg$ capacity it needs $A_1=0.31mm^2$ cross-section at the bottom. At 1000mm width that's 0.00031mm thick.
Now I'm really not in the mood to solve a differential equation of thickness - mass - tensile strength - gravity so let me try a discretization, approximating with $h=1km$ long wedges. At 35000 samples that should give us a decent approximation.
$$
V_n= {A_n+A_{n+1} \over 2}h \\
M_n=\rho V = \rho {A_n+A_{n+1} \over 2}h
$$
Now we can't happily assume weight not to vary with altitude. After all, near the orbit it will be zero. It varies with distance from center of Earth. At surface, $r_0=6378km; M_{earth}=5.97 10^{24}kg; G =6.67300 × 10^{-11} {m^3 \over kg s^2}$; 
So, the weight function of each segment will be 
$$
Fw_n=G{M_n M_{earth} \over r_n^2} \\
r_n=r_0+n[km]
$$
And the tensile strength surface $A_{n+1}$ must overcome is 
$$
F_{n+1}=F_n + Fw_n \\
F_{n+1} = A_{n+1} UTS 
$$
We seek $A_{35000}$ which will trivially yield thickness by dividing by 1000mm.
$$
A_{n+1} UTS = F_n + Fw_n \\
A_{n+1} UTS = F_n + G{M_n M_{earth} \over r_n^2} \\
A_{n+1} UTS = F_n + G{\rho {A_n+A_{n+1} \over 2} h M_{earth} \over r_n^2} \\
A_{n+1}  = F_n + (A_n+A_{n+1}){ G \rho h M_{earth} \over 2r_n^2 UTS } \\
X := { G \rho h M_{earth} \over 2r_n^2 UTS }\\
A_{n+1} -  = F_n + (A_n+A_{n+1})X \\
A_{n+1} = F_n + X A_n + X A_{n+1} \\
A_{n+1} - X A_{n+1} = F_n + X A_n \\
(1-X)A_{n+1} = F_n + X A_n \\
$$
We get our two fundamental equations for numeric computation: (with helper X, which I'm really not in the mood to transform into something nicer.)
$$
X = { G \rho h M_{earth} \over 2r_n^2 UTS }\\
A_{n+1} = { F_n + X A_n \over (1-X) } \\
F_{n+1} = A_{n+1} UTS 
$$
Now excuse me, it's 3AM and I'll finish the calculations at a different time.
A: The level at which this question is being asked is uncertain.  I thought I would mention the idea of the space elevator, which some people take seriously.  However, Lubos is correct in saying that the edge of the atmosphere does not mean the end of gravity.  A spacecraft orbits the Earth because it is falling towards the Earth.  However, it is just moving fast enough so that it keeps missing the Earth which curves away under the spacecraft’s trajectory.  The apparent loss of gravity, as seen with space shuttle and ISS astronauts floating around, is due to the fact the astronauts and everything in the spacecraft is falling and moving with the rest of the spacecraft.  Remember that Galileo demonstrated that different masses fall with the same acceleration, and so everything in a spacecraft falls with the same acceleration.  However, the whole thing is moving fast enough to keep missing the Earth, and this acceleration of gravity provides the centripetal force which maintains a circular orbit.
There is this “Jack and the Beanstalk” idea of the space elevator.  I seriously question whether this will ever be built, but the idea is possible in principle.
http://en.wikipedia.org/wiki/Space_elevator
The idea has some problems of course.  In particular it is tough to stack up mass elements without it falling over.  If the gravity force at the center of mass deviates from its foundation the stack fall over.  So I think the idea of building the tower from the ground up is probably wrong.
The prospect for this lies in building from the top down.  There are ideas about manipulating the orbits of asteroids.  The Russians want to change the orbit of Apophis asteroid, which will come close to the Earth in 2029.  Suppose we get good at doing this, and we manipulate the orbit of an asteroid into geosynchronous orbit around the Earth.  A geosynchronous orbits is at a radius of 37,000 km where the orbital period is equal to the rotational period of the Earth.  As the Wiki page shows one must then have a counter weight beyond geosynchronous orbital radius.  So if one had an asteroid of sufficient mass and with the proper material constitution one could then build the tower downwards from this point.  This would be accompanied by building upwards with an amount of mass so the center of mass of the emerging structure remains at geosynchronous orbit.  Eventually this would then be constructed into this tower.  The gravity gradient on this emerging structure would have to be carefully monitored and the vibrations on this controlled.  It would not be at all trivial to do this.
A: There's an interesting review article on the subject:
Review of New Concepts, Ideas and Innovations in Space Towers
Mark Krinker, (2010)

A lot of new concepts, ideas and
  innovation in space towers were
  offered, developed and researched in
  last years especially after 2000. For
  example: optimal solid space towers,
  inflatable space towers (include
  optimal space tower), circle and
  centrifugal space towers, kinetic
  space towers, electrostatic space
  towers, electromagnetic space towers,
  and so on. Given review shortly
  summarizes the[s]e researches and gives
  a brief description [of] them, note some [of]
  their main advantages, shortcomings,
  defects and limitations.

http://arxiv.org/abs/1002.2405
The above is sufficiently interesting that I would recommend it as a place to look for interesting problems for your undergraduate classes in mechanics.
A: Re: the original question:
If you wanted "zero gravity" at the top of the tower, you'd have to build a tower tall enough to reach the height of geostationary orbit: a point at which the orbital period of an object in freefall matches the time it takes the earth to rotate once.  As other commenters have pointed out, that'll occur at a height of roughly 35000 km above ground.  Good luck!
Re: The claim that "but one may still say that the limitations are of an engineering (and budgetary) character rather than fundamental physics limitations."  I disagree.  One fundamental physical limitation is the fact that matter is held together by chemical bonds.  This limits the ultimate strength of any material (although AFAIK for all macroscopic materials mankind currently produces, the ultimate strength is far shy of what one would get from a "perfect" material).  The ultimate strength will limit how tall a tower one can build.  This is, for example, the reason why small asteroids can be quite aspherical, but large asteroids (and planets) cannot: an highly aspherical planet would collapse under its own gravitational force and become near-spherical due to the finite strength of the materials it's made of.  The ultimate strength limit also means that you can't build the "space elevator" other folks mention using any currently available materials that I know of.
A: First of all, it is an elementary misconception that there would be a "zero gravity" environment in a tower that would only reach the top of the atmosphere.
Most of the air molecules exist at a height smaller than 10 kilometers - and above 100 kilometers from the Earth's surface, the air is so diluted that it becomes undetectable.
At the height of 10 kilometers - where the atmospheric pressure is almost zero - the gravitational acceleration is just 0.3% weaker than it is on the surface and even at 100 kilometers, it is just 3% weaker. So forget about "lunar games". The gravitational forces over there are pretty much indistinguishable by humans from those we know on the surface. At 100 kilometers, a 75-kilogram person may feel 2 kg lighter but it may be compensated by the suit he needs to avoid suffocation. ;-)
The absence of air has nothing to do with the absence of the gravitational force. The air tries to be at low attitudes in order to minimize its potential energy; how much it wants to minimize the energy is given by the molecular mass and the temperature. However, the air density is something totally different than the gravitational acceleration - they're surely not proportional to each other in any sense. 
The air density is proportional to $\exp(-\Phi m / kT)$ where $\Phi$ is the gravitational potential, $m$ is the molecule's mass, $k$ is Boltzmann's constant, and $T$ is temperature in Kelvin's degrees. However, the gravitational acceleration is $d\Phi / dh$. These two functions depend totally differently on the height $h$.
Tall buildings
The tallest building in the world is Burj Khalifa in Dubai - it has 828 meters. It's about 10% of the thickness of the atmosphere in the "narrow sense". It's hard to build tall buildings - one must guarantee that they're stable despite the immense weight of the material above each floor and despite the wind and vibrations of the Earth's surface.
But there are no "strictly physical" limitations that would prevent one from building a tower that reaches 10 kilometers above the surface. One may say that all such limitations are of engineering character. Tall mountains such as Mount Everest may be viewed as "natural tall buildings" and their height isn't far from the top of the atmosphere (in the narrow sense). The design of very tall buildings would probably have to be a bit hierarchical - with a solid base made out of a heavier material and lighter floors near the top, just like in the case of mountains.
One would surely start to face problems to find reasonable materials if he wanted buildings that substantially reduce the gravity on the roof - buildings that are thousands of kilometers tall. For example, one of my kindergarten visions was to build an elevator that could take one to the Moon and that could convert the kinetic energy of the Moon's motion  around the Earth. That's a really challenging task for engineers. One will run into problems with conventional materials etc. - but one may still say that the limitations are of an engineering (and budgetary) character rather than fundamental physics limitations.
A: Towers supported from the bottom are a bit tricky.
Buckling
  limits how tall a column can be. One needs to additional lateral stiffness to overcome this, usually by putting up guy wires. Even so there are going to be real limits, as Anonymous Coward has mentioned above, solids obtain their stiffness from chemical interactions between molecules and atoms, and the strength to weight ratio is limited. There are some plans for some structures up to about a kilometer, but the cost per unit volume of building goes up for tall buildings. We could probably go a lot higher by the use of carbon nanotubes, but we are years away from being about to construct practical guy wires from them.
A: Rather than using material, perhaps magnetic fields configured in stages. Imagine a stack of plates separated at a distance on the order of a meter. Magnetic fields, from superconducting magnetics repeal the plates above or below. Sensors and an electronic system dynamically adjust the fields. I wonder if the fields would need to become attractive at some point due to centifigual forces from the earths rotation. 
