Consider a photon born on Earth, and traveling to a distant galaxy light years away. In the reference frame of the photon, the distance between Earth and the galaxy is contracted from Earth's $L_0$ to
$L = L_0 \sqrt{1 - \frac{v^2}{c^2}} \longrightarrow 0$
The time required to get to the distant galaxy is also "contracted," relative to our time $T_0$, as
$T = T_0{\sqrt{1-\frac{v^2}{c^2}}} \longrightarrow 0$
However, if we compute the limit of $\frac{L}{T}$ instead as $v \to c$, we get
$lim_{ v \to c}\frac{L}{T} = lim_{ v \to c} \frac{L_0}{T_0}=c$
The speed of light in a reference frame moving at the speed of light is still the speed of light, but it becomes rather meaningless since light does not live in the time domain. We often say light lives in only spatial domain, but according to equations it appears it does not travel through space in its own reference frame either. In order to measure speed, we need length and duration - i.e. travel. Since neither length nor time appears to progress from the photon's perspective - i.e. it is born and ceases to exist in the same moment, and in the same place : everything in between is contracted to a single point in the direction of travel - then there is no speed.