Could a human beat light in a footrace? Is there anything preventing the following experiment from being done right now?

Imagine that a human ran from point 'a' to point 'b' while light
  particles that reflected off a clock moved through a special medium
  from point 'a' to point 'b' as well. Could a human arrive at point 'b'
  before light? (As for the special medium, I'm imagining something like
  a complex maze of mirrors submerged in a very dense material.)

If so, if the human waited at the finish line to view light arriving in second place, would they see the time that the race began on the clock?

 A: No physical laws are being broken in this thought experiment. If you are concerned with the relativistic requirement "nothing can go faster than the speed of light", that only applies to the speed light goes in a vacuum: $c = 3 \times 10^8$ m/s. The reference to light in that relativity postulate makes it sound like if you could only find a situation where you slowed light down, you could break the laws of physics; not so. A better statement of the postulate would be "nothing can go faster than $3 \times 10^8$ m/s, which happens to also be the speed light travels at in a vacuum." I don't see anyone going faster than $3 \times 10^8$ m/s in this thought experiment, so no physics violations.
As for what the human at the end of the race sees:
He sees a blinding blue light from the all the Cherenkov Radiation from even the slightest charged particle passing through the medium. And perhaps the time at the start of the race. It's exactly what you would imagine since we are talking non-relativistic speeds. What an anti-climactic answer, eh?
A: It is of course possible not just for a human, but for anything "moving" to beat light moving in a given medium. It is impossible to beat $c=3\cdot10^8\ \mathrm{m/s}$, but not the velocity of light in a given medium.
In the experiment proposed, the man arriving at the end will see time exactly the same way a man out observing, it will not change the notion of Newtonian time, actually.
There is a (natural) misunderstanding about the light velocity (or the velocity of electromagnetic waves): it is always $c=3\cdot10^8\ \mathrm{m/s}$. 
What happens in other media, is that light propagating around atoms/molecules is constantly absorbed and re-emitted for the electrons given the effect of "slowing down" the net velocity of the light in that medium. But the light velocity, around the medium continues to be $c=3\cdot10^8\ \mathrm{m/s}$, for the effect of movement and relativity theory (except for the absorption and emission moments).
A: There is a concept of "slow light" which is looking at light pulses whose group velocity propagates very slowly.  This is slightly different than your clock example, but close enough that you might be interested in it.

In 1998, Danish physicist Lene Vestergaard Hau led a combined team from Harvard University and the Rowland Institute for Science which succeeded in slowing a beam of light to about 17 meters per second...

Usain Bolt can run roughly 12m/s, so we are not all that far off.
(Of course, we are playing some tremendous games with the light beams while doing these sorts of slow light games.  Whether this is actually applicable to your specific thought experiment involving a clock depends on what aspects of the experiment you felt were important)
A: The dense material would diffuse the light so if you got it dense enough you probably could not read the clock.  
In theory if you had enough perfect mirrors you could see the start time.
A: To focus on just one aspect of the answer:

If so, if the human waited at the finish line to view light arriving in second place, would they see the time that the race began on the clock?

The medium would have to do the majority of slowing down - after too many mirrors you increase the distance such that the clock appears too small to be legible.  If you used slightly convex mirrors to magnify it you'd end up shooting nearly all the light outside the medium, and it would be too dim to see.  If you increased the brightness to accommodate that loss, you'd vaporize everything within the initial beam of light.
So while the rest of your thought experiment doesn't pose a problem, there are certain physics limitations which would make it so the runner couldn't see the clock from the past.
A: Yes, it is theoretically possible. 
For example, you could use two parallel, perfectly reflecting mirrors of length $L$, where $L$ is the distance between point A and point B. Let the distance between the two mirrors be $d$.
Assuming that the ray of light enters the two mirror by hitting one of them close to point A at an angle $\theta$, it will be reflected
$$N \approx \left ( \frac{L}{d \tan \theta} \right)$$
times before arriving at point B, covering a distance
$$l = N \frac d {\cos \theta} \approx \left ( \frac{L}{d \tan \theta} \right) \frac d {\cos \theta} = \frac L {\sin \theta}$$
in the process. Notice that the result does not (somewhat surprisingly) depend on $d$. 
Therefore, the time needed to go from point A to point B for the ray of light is
$$t=\frac l c \approx \frac L {c \sin \theta}$$
You can therefore define an "effective speed" $v_e$ for the ray,
$$v_e \equiv c \sin \theta$$
If the speed of a human is $v_h$, the human will be faster than the light ray if
$$v_h > v_e \ \Rightarrow \ \sin \theta < \frac{v_h} c$$
The record speed for a running human (*) is  $44.72$ km/h (Usain Bolt, 2009). The speed of light in a vacuum is $1.08 \cdot 10^9$ km/h. You get therefore the condition
$$\sin \theta < 4.14 \cdot 10^{-8}$$
You can see therefore that this is not very easy to realize in practice (and we are neglecting refraction, absorption, scattering, surface roughness etc.).  
You also can repeat the calculation assuming that there is a material with refractive index $n$ between the two mirrors.
(*) I want to be generous.
A: I identify 3 questions.  The answers are:
First is yes, a human could beat "light" in a timed competition, if the "right" conditions are met. If the light is deflected and sent on a "long enough" path, the light will arrive latter than the human at the finish line.
Second is also yes, if instead of the clock face, a flash of laser light is used, then it is doable (see wikipedia Lunar Laser Ranging experiment).
And Third, if a 1 second "flash" is used (instead of clock face), then the human will see a 1 second flash, some time after getting to the finish line.
For an example, lets use: a 10m distance from start to finish, the human can run 10 m/s, and the laser flash is "bounced" off the moon. 
The human starts running when he sees the 1 second flash. In 1 second he will get to the finish line (t=d/v) and 1.5 seconds later (2.5 - 1), he will see the "same" 1 second flash again. 
