Why can, or can not, a perfectly incompressible fluid exist? Water is normally assumed to be an incompressible fluid - for example in the context of calculations involving water pressure.
I wondered whether that is strictly true, or an approximation? Later I noticed some side note implying water is not fully incompressible.
Of course that makes sense, as there are not many things "perfect" in nature in this sense, like maybe supraconductivity and suprafluidity.

Now, why, or "in what kind of way", is water only almost incompressible - is it caused by impurities like gases and other fluids in the water?   
Or is it not completely incompressible in some fundamental way?
 A: Matter is made up from point like fundamental particles, like electrons and quarks, that have zero volume. This puts us in the interesting position where the true volume of all matter is zero, and the only reason that everything doesn't instantly collapse into a point of zero volume is that the pointlike fundamental particles maintain a finite distance from each other due to a variety of forces. For example in a hydrogen atom the uncertainty principle restricts how close together the electron and proton can get. You can compress a hydrogen atom, but it costs energy so there will be a repulsive force resisting the compression. Similarly if you try and squeeze two hydrogen atoms together the exchange force resists you.
The point of this rather abstruse discussion is that for a system to be incompressible the force between pairs of fundamental particles would have to become infinite. For any finite force the particles can be squeezed together, and therefore the system can be compressed. No such infinite forces are known in nature, and therefore there is no such thing as a fundamentally incompressible system.
The compressibility of macroscopic systems like water is the reciprocal of the bulk modulus, which for water is about 2.2 GPa. This is high compared to readily compressible systems like gases, but low compared to steel at 160 Gpa and diamond at 443 GPa. So compared to steel and diamond water is actually quite compressible. 
I think diamond has the highest known bulk modulus of normal solids. I'd imagine more esoteric states of matter like degenerate matter would have a (much) higher bulk modulus, but it would still not be incompressible.
A: Formally, the incompressibility of a fluid is defined by the compressibility,
$$
\beta=\frac1\rho\,\frac{\partial\rho}{\partial p}
$$
where $\rho$ is the mass density and $p$ the gas pressure. This means that, the compressibility is the measure of how much the density (volume) changes when a pressure is applied. 
For water at standard pressure, this works out to be on the order $10^{-10}\,m^2/N$ which is pretty darn small but definitely non-zero. This value isn't due to impurities in the water, it is due to the properties of H$_2$0 itself. If you look at the $\rho\,{\rm vs}\,p$ plot, you can see how the density changes with both (solid green line):

The left-most end of the chart is standard pressure of 1 atm (roughly 0.1 MPa). As you can see, it is not until very high pressures that the density begins to really change. But at most every-day temperatures the deviations from horizontal are negligible.
A: Water in its liquid form is almost incompressible because of the tendency of H- bond not to reduce in length after a certain limit. This can be said in accordance to Hydrogen atom as well. For example,no matter how much you compress the atom, its size will never reduce.
