Why is the speed of light arbitrarily the limit? I know Einstein was great and all. Why is it that exactly at the speed of light is where infinite energy is required to accelerate any object with mass? Is it simply because the math of relativity checks out and explains most of everything? Are there any physicists who disagree with Einstein's theory? 
 A: Let's assume, for argument's sake, that the Galilean transformation holds rather than the Lorentz transformation.
Then your questions would become

Why is infinite speed arbitrarily the limit?  Why is it that exactly
  at infinite speed is where infinite energy is required to accelerate
  any object with mass?

I suspect that you wouldn't, in fact, think of asking such questions since they almost answer themselves.
Moreover, infinite speed would be an invariant speed - a speed that is measured to be the same in all reference frames - since (loosely speaking) $\infty + v = \infty$.
If we ask the question "what if there is a finite invariant speed", the mathematical answer is the Lorentz transformation where $c$ is the finite invariant speed.
Indeed, if we let $c \rightarrow \infty$ in the Lorentz transformation, we recover the Galilean transformation.
From this perspective, the result that accelerating to the invariant speed requires infinite energy and is thus impossible, doesn't seem so odd.
To summarize, without postulating that the speed of light is invariant, one can derive the form of the Lorentz transformations from just the principle of relativity.  In this form, there is an undetermined finite invariant speed.
That light propagates at the invariant speed is then simply an empirical fact rather than a postulate.
A: 
Why is it that exactly at the speed of light is where infinite energy
  is required to accelerate any object with mass? Is it simply because
  the math of relativity checks out and explains most of everything?

To say that the math checks out is the wrong way of putting what's going on because it separates the math and the explanation for the math. Einstein had explanations for why he picked the equations he picked. That is, he had an account of what was happening in reality to bring those outcomes about and without that explanation the math wouldn't have any relevance to physics. Einstein's explanation involved ideas about what kinds of measurements are allowed by the laws of physics, see:
http://www.fourmilab.ch/etexts/einstein/specrel/specrel.pdf

Are there any physicists who disagree with Einstein's theory?

General relativity is the best available explanation of gravity and the structure of spacetime. There is a problem with the theory in the sense that it is not a quantum theory of gravity. There is no reason to think that its replacement will involve going back to anything that resembles Newtonian mechanics.
A: It's not actually. It's not like light has some special status in the Universe. It's just that there is a maximum speed, and light, among other things, tends to get very close to that limit. In practice light is slowed down by its environment, and there was even speculation at one point, that light mad a bit of mass and so neutrinos might move a tiny bit faster. This was retracted, but it goes to show that the thought is possible.
A: For the non-physicists amongst us, Brian Greene gives a non-mathematical but intuitively satisfying explanation in 'The Fabric of the Cosmos': In a loose sense, when we are at rest, we are moving through time at the speed of light. As we start to move through space, what we are actually doing is diverting more and more of that motion to a spatial direction rather than a time direction, but the combined velocity remains constant. Thus we cannot exceed the speed of light, because at that point, we have diverted the entirety of our motion to the space direction.
(I'm paraphrasing, but I believe I've caught the gist)
A: 
Is it simply because the math of relativity checks out and explains most of everything?

Yes, the equations of special relativity have been tested against data innumerable times and have never been falsified.

Are there any physicists who disagree with Einstein's theory? 

As far as I know, no.
