Q:

How empty of fuel are spacecraft booster rockets typically?

A recent XKCD What-if article mentions the situation where each additional kilogram of cargo to LEO requires an additional 1. 3 kilograms of fuel, which in turn requires fuel to carry (simplification). I wonder then, if an additional half-kilo object, such as an iPod will have a large impact on the amount of fuel that engineers will load into the ET. Is the ET, and other rockets, always 'topped off' regardless of payload, or is the amount of fuel loaded carefully measured? What tolerances are acceptable? How much excess fuel is typically in the rocket / tank when the payload is separated?

A:

Consider Tsiolkovsky's rocket equation
$
\Delta v = v_e \ln \left( m_0/m_f \right)
$
with $\Delta V$ the total change in velocity, $v_e$ the exhaust speed of the reaction products, $m_0$ the initial mass (structure+payload+propellant) and $m_f$ the final mass (structure+payload). 
If you ignore the atmosphere and other such "nuisances", it should be obvious that for any given rocket $\Delta V$ is a value independent of the rocket's mass -- the rocket needs to be given a certain $\Delta V$ to reach space. Also, $v_e$ is a value that depends mostly on engine specifics, and is thus fixed for a specific type of rocket. It follows that $\Delta V = C_1$, a constant, and $v_e = C_2$, another constant. 
You can then rearrange Tsiolkovsky's rocket equation to
$
 m_f = m_0 e^{-\Delta v/v_e} = m_0 e^{-C}
$
where simply $C = C_1/C_2$. Now, if some extra mass $a$ were to be added to the rocket's payload, this would translate into
$
 m_f + a = a + m_o e^{-C} = \left( a e^{C} + m_0\right)e^{-C}
$
in other words, it would require an additional $a e^{C}$ kilograms of propellant (note that I'm avoiding the word fuel, as that is generally used to describe a substance that reacts with oxygen, which isn't necessarily the case for rockets). The term $e^{C}$ is commonly called the growth factor. 
As a numerical example, suppose 
$
\matrix{\Delta V = 15\ \mathrm{km/s}\\ v_e = 7\ \mathrm{km/s}}
$
which is roughly the maximum capability of today's technology (and yes, ion engines are much better in this respect, but these can not be used for manned launches, so we'll not consider them here). Then 
$
 e^C = e^{15/7}\approx 8.5
$
so you'll need approximately 8.5 kg extra propellant for each added kilogram of payload. 
"Topping off" the rocket as you suggested can be rephrased as adding propellant mass to payload mass, which has possibly severe penalties on the maximum deliverable payload mass. Although a safely factor of around 1.1 is usually employed, rocket fuel is expensive, so "topping it off" is usually not done. 
Moreover: engines running on solid propellants (as in booster rockets) can in general not be turned off, which would make topping off a complete waste as boosters are usually discarded to save on final structural mass. Also, many non-booster rocket stages are discarded mid-flight for the same reason, so topping off would only ever make sense in the final stage, provided that that stage is part of the payload and therefore not discarded.  
So in summary: aside from a small safety factor, launch vehicles are filled with an amount of fuel specifically tailored to the mission. 

