Is the elevator analogy of the equivalence principle really true? The equivalence principle as I understand it goes something like this:
Let's suppose you're in a black box in the middle of nowhere in space, and we accelerate this black box in some direction. You'd feel a force just like you would if you were in a gravitational field which was causing the same acceleration.
In fact there is no experiment that you can do to say for sure if you are in a gravitational field or just accelerating in some direction in a black box.
OK, so now let's imagine this elevator is being accelerated. Surely it can't accelerate at a constant rate forever, right? It's limited by the speed of light. So at some point or another the elevator will start to slow down. This sort of deceleration just does not happen in a gravitational field.
Where did I go wrong?
 A: To begin with the equivalence principle claims that there is no difference between standing in a gravitational field and constant acceleration in deep space. This is only almost correct. The correction being that real gravitational fields have noticeable tidal forces (This is what made Einstein think of Curvature causing Gravitation). So if you were indeed in a real gravitational field, you can feel the tidal forces (if you are sensitive enough) stretching and squeezing you, while no form of constant acceleration can really make you feel tidal forces. However it must be noted that when Einstein gave the equivalence principle he clearly mentions that it is true locally, which means that there is no measurable tidal forces. But if you try to generalise the principle to larger scales, the ep is not valid.
Again by acceleration we mean four-vector acceleration, which is defined as 
$$a^\mu = \frac{d^2X^\mu}{d\tau^2}$$
Where $\tau$ is the proper time. This gives us the relativistic version of acceleration, hope that clears your doubt about acceleration as well.
A: The point of the thought experiment isn't to say that the elevator can accelerate forever. The point is that acceleration is indistinguishable from being in a gravitational field. The acceleration doesn't have to exist for all time. However, just to address another issue...

It's limited by the speed of light. So at some point or another the elevator will start to slow down.

This is incorrect. The speed of light limit doesn't say you will start to slow down once you get closer to the speed of light. The issue is that you are thinking in terms of absolute velocity, not relative velocity. The elevator cannot accelerate to move faster than the speed of light relative to something else. This doesn't affect the acceleration you would feel in the elevator though. If we had some magical infinite fuel source, then indeed you would feel the same acceleration in the elevator forever. The speed limit applies to an outside observer who would see your speed relative to them approach but not reach the speed of light.
A: The thought experiment is to show that gravity and acceleration are equal. It has nothing to do with how long an acceleration can be sustained.  If that’s all your worried about just imagine the elevator swinging in a large arc where the centripetal force creates artificial gravity. 
A: Think about it like this: in normal Special Relativity, even when you are going at the speed of light, you will not know it. You can say that you are at rest, irrespective of your speed.
It is the same in this situation: you are inside an elevator and can't look outside; and you just feel that you are being pulled by the floor, so you conclude that you are standing in a gravitational field. As @BioPhysicist wrote in his answer, someone outside would see you approach lightspeed, but that does not mean that you would notice or even that you slow down.
Let me give you an example: suppose a person going at near lightspeed in his spaceship passes the Earth. He is travelling vertically (he first passes the North pole and then the South Pole). So, he says that he is at rest nd it is the Earth that is moving away from him at near lightspeed. But does that actually affect you, standing on the Earth in anyway? No, because in your point of view, you are being pulled by the Earth's gravity.
Say, the person in the spaceship tries to measure your acceleration relative to the Earth. He will find that you are still being pulled at the same acceleration. That is the key point here: your acceleration with respect to the elevator will never change, just because some outsider says that you are approaching lightspeed.
That is what the Equivalence Principle says. Your acceleration relative to the elevator is indistinguishable from a gravitational field.
A: You can think of it in the following terms-
As you approach closer to the speed of light, to produce say a change $\Delta v$ in the velocity, you would need to pour in more energy to the system than you would have to at low velocities. To see this, calculate the amount of energy that you will need to produce a small change $\Delta v$ in the velocity. Under binomial approximation, you get that $\Delta E = ma^3v \Delta v$, where $a$ is the Lorentz factor, and $m$ is the rest mass of the system.
From here you can see that at high velocities, you need to pump in more energy to the system to produce the same change in velocity compared to what you would have to at low velocities. You can check that as one goes closer to the speed of light, $\Delta E$ blows up.
So the problem is actually that you will need an infinite amount of energy to reach the speed of light, or in other words, you just can't reach the speed of light!
