Wouldn't a photon disappear because of length contraction? I was experimenting with the formula for length contraction, when I realized that anything traveling at the speed of light shrinks out of existence. This is the formula for length contraction:
$$T=T'\sqrt{1-\frac{v^2}{c^2}}$$
Where $T$ is the observed length and $T'$ is the proper length. When the speed of light, $c$, is plugged in for $v$, then the formula simplifies to this:
$$T=T'\sqrt{1-1}$$
Which further simplifies to this:
$$T=T'(0)$$
Therefore, anything traveling at the speed of light will not be seen, regardless of actual length. So, wouldn't we not be able to see any particle traveling at the speed of light? 
 A: I am only a layman, so don't take this answer seriously.
This length contraction formula, and the whole Special Relativity in its original form, is for macro-sized, non-quantummechanical objects. Thus, the formulas work if you want to calculate the size of a spaceship nearing the speed of the light. And not if you want to calculate the size of photons with exactly the speed of the light.
Photons are elementary particles. Or waves. Or both of them. They are quantummechanical objects, described by QED (quantumelectrodynamics), which is a field theory.
So, the theories seem so:
      1. Classical (newtonian, non-QM) mechanics
             /                   \
2. Quantummechanics            3. Special relativity
               \                /
      4. Relativistic Quantummechanics (QED, QFT)

You are now thinking in (3), but your problem is answered by (4).
Essentially, photons are the waves of the electromagnetic field, and these waves propagate with c.
Unfortunately, QFT hasn't so simple and beautiful formulas as the SR, but it is tremendous fun even trying to understand them.
Extension: There is also a possibility, that you are calculating only with classical electrodynamics, without any QM. But there are no photons in classical electrodynamics, there is only the electromagnetic field described by the Maxwell Equations. Here it is possible to calculate light as a wave packet. In this case, you don't have any particles, there is only a propagating wave of the EM field.
A: Anything moving at speed of light loses its reference system. This is the reason why your calculation yields strange results. Everything is multiplied by zero so that not only photons (which indeed have no length) but also long distances are reduced to zero.
