No, you would not get the block to the speed of light.  The situation you describe is fine for classical-physics situations, but you can't apply classical physics to relativistic problems. 

The boxes would indeed speed up in the manner you describe up to about 10% of the speed of light.  Above this, relativistic effects begin to dominate.  The energy being supplied to the block will manifest as extra mass - the block will appear heavier, and therefore more energy is needed to accelerate it further.  The more it accelerates, the heavier it gets, until it needs an infinite amount of energy to push it to light-speed. 

The same reasoning applies to each of the conveyor belts - each belt moving on top of another belt would be heavier than the slower ones beneath it, and would need more energy to propel it forward.

matter reaches infintite mass once it hits the light barrier arcording to einstien thats why you cant travel faster than the speed of light its not fast enough anyway what you want is a hyper drive or worm hole!

Exceeding the speed of light?  No sweat!  All you need to do is use a "quantum conveyor belt".  The well established theory of "quantum entanglement" provides that once you observe the state of a quantum particle it will have a corresponding effect on another particle at a different location.  If the other particle is at such a distance that it forms what is called a "quantum non-local connection" then the transfer of "information" that determines the state of the second particle occurs at a speed not less than 10,000 times the speed of light, according to a 2008 experiment with findings published in a prestigious scientific journal.  Einstein referred to this phenomenon as "spooky action at a distance" so the theory is well known and further research now being conducted into quantum coupling may eventually result in many practical applications for transmitting and processing information.

The answer actually is that velocities don't just add up.  You have to study relativity to understand why (and is was a long time ago that I, sort of, understood).  At every day speeds just adding together is accurate enough  - you are only out by an amount equal to the product of the velocities divided by the square of the speed of light.

Just because the conveyor belt is a practical impossibility - due to infinite mass - doesn't really answer the question.  The counter-intuitive idea that velocities don't simply add together does.

My personal variant of this question is consider a long, stiff stick.  I push one end, doesn't the other end move instantaneously, therefore faster than light transfer of information?  The answer is, it depends what you mean by stiff.  Infinitely stiff would do the trick, but there you go ... no infinitely stiff stuff exists so real sticks move like a spring. 

look at the work been done on slowing light and how light travells trough spacetime and youll have youre answer