Is it possible to have a planet entirely made out of liquid water? Earth is mostly covered in oceans, but they only go a few kilometres deep. It's obviously not possible to have a planet the size of the earth to be made entirely out of water, because of the kind of pressures reached in the interior.
a. But say that we did, how far down from the surface would water remain water before presumably turning to ice under the pressure?
b. How large a 'planet' could we have made entirely out of water? Would it be able to attain the size of a small dwarf planet like Ceres?
 A: Suppose we assemble a large mass of water in space, then it will form a sphere held together by its gravitational field. The question is then how large this sphere can become before the pressure at the centre causes the water to solidify to ice.
The calculation of the pressure at the centre is straightforward in principle, and is described in How to find the force of the compression at the core of a planet? The problem is that to do the calculation precisely we need to know how the density of water changes with pressure. There is no simple equation for this so we would need to do a numerical calculation. However if we make the approximation that the density of the water remains constant we can get an approximate equation for the pressure:
$$ P = \frac{2}{3}\pi G \rho ^2 R^2 \tag{1} $$
We can estimate the pressure at which water solidifies by looking at the phase diagram of water. The following phase diagram comes from London South Bank University web site:

The pressure at which the water solidifies to ice is strongly temperature dependent. At everyday temperatures it's around 800MPa to a GPa, and since this is an approximate calculation let's take a GPa as being a round number. Then using equation (1) we find that value of the radius $R$ for which the pressure reaches 1GPa is about 2700km.
So there's your answer. A ball of water at room temperature larger than 2700km would contain a solid ice core. In practice the density of the water increases with depth so the radius at which ice forms would be less than this. Though the relative density of ice VI is only around $1.3$ so it wouldn't be that much smaller.
A: The related question points out that water would become ice at a depth of around sixty km. This answers the first question. And this suggests we should expect a body of water a 120km in diameter to remain water all the way through. This is far smaller than a dwarf planet and more the size of a large asteroid.
