Does the pilot of a rocket ship experience an asymptotic approach to the speed of light? A question has recently come up that goes beyond my knowledge of special relativity.  Suppose a pilot has his foot on the gas pedal of a rocket ship and keeps it applied to achieve a constant acceleration, and he has a magical engine that can maintain the thrust needed to accomplish such acceleration even as the vehicle acquires mass.  He measures his acceleration at periodic intervals and sums it up as he goes.
Observers perceive the rocket to increase in speed and gradually approach the speed of light (but never reach it). Two possibilities are being debated for what the pilot observes, and my preference is for the first:


*

*Time dilation gives him the impression that the summation of his readings are approaching light speed at a constant rate, but there is an asymptote involved here even in the inertial frame such that he never actually perceives himself to reach, let alone exceed, $c$, because time slows down to a crawl (as seen from the reference frame) before this can happen.

*Although to observers he is only approaching $c$, it is possible for the summation of the measurements on his accelerometer to indicate that he has surpassed $c$ and is now traveling faster than light.  From the reference frame he is approaching $c$ and his instruments in the inertial frame tell him that he has surpassed it.


I assume (1) is true beyond a doubt?  If so, is there a straightforward explanation for this?
 A: The equations you need are given in John Baez's article on the relativistic rocket. If $T$ is the time measured on the rocket, and $V$ is the velocity measured on the rocket, then the equation you need is:
$$ V = c \tanh \left( \frac{aT}{c} \right ) $$
For a rocket accelerating at $g$ the velocity time looks like:

So, as you guessed, option (1) is the correct answer.
Bear in mind that the crew of the rocket assume they are stationary and that the rest of the universe is accelerating towards them. Their situation is as if they were stationary in a constant gravitational field of $g$ and were observing everything else falling towards them in that field. It should come as no surprise that they do not observe any superluminal velocities.
If you're interested in taking this further, the coordinate system for a uniformly accelerating observer is called the Rindler coordinates.
Incidentally, one of the few things observers in two frames will agree on is their relative speed (measured as the two observers meet i.e. when they're at the same point in space). You'll find that the time dilation and length contraction balance each other out to give both frames the same relative speed.
A: This question intrigued me enough to spend a few hours on it.  I like to address special relativity problems in more than one way, so this gets to the same answer as John's but with different logic.
The problem with the initial concept is that we are assuming v = at, as Newton would say.  If the acceleration is 9.81 m/s2, then sure enough, after 1 year (31e6 seconds), we would calculate a velocity greater than light speed.  Why does this not work?
One way to demonstrate it is to use a conservation of energy approach.  As you know, Newton would say kinetic energy is 1/2mv^2 while in relativity, we show that is only true for velocities well below light speed.  We would say if we use enough force to create an apparent acceleration of 9.8 m/s2 that:
\begin{alignat}{7}
\int{ F ds }           & ~=~ (\gamma - 1)mc^2
\end{alignat}
Plug in that
\begin{alignat}{7}
F           & ~=~ \frac{d(mv)}{dt}
\end{alignat}
and acceleration a is dv/dt, you can eventually show that
\begin{alignat}{7}
v           & ~=~ \frac{1}{\gamma^3}at
\end{alignat}
which reduces to the Newtonian result when velocity is modest.  I am not sure if that is a well known equation, but it solves this problem nicely.  Plus you see that the underlying cause is that the acceleration (which is energy addition) is not going to velocity increase, but rather to momentum increase.  And momentum can go to infinity without the velocity reaching lightspeed.
The formula also explains why the pilot doesn't calculate greater than light speed by "the summation of the measurements on his accelerometer."  He should not increment his speed  by at.  Instead, his calculation will be, solving numerically, the previous total calculated velocity plus over the next time period, a reduced acceleration, so that light speed is approached asymptotically.  I did a spreadsheet and it looks like John's result:

A: This question can be also answered effectively using Newtonian mechanics. As you approach continuously accelerating at a constant rate, say with a=1G acceleration (same as Earth gravity so that you sustain no biological damage from prolonged acceleration), aiming towards the speed of light
is effectively the same as if the spacecraft mass inertia and everything everyone’s mass that is in there, is continuously increasing F=ma.
Which means that you propulsion needs to produce more and more force F thus to consume more and more energy in order to sustain the total mass of the spacecraft m, at a=1G acceleration, that is 1G=9.806 m/s^2. This amount of energy and force needed becomes infinite as you go 100% of the speed of light. Infinite in this case means impossible.
This is because fermion particles like electrons and quarks that consist your body and actually every condensed matter, are mass particles which "refuse" to travel in space at speed 100% the speed of light c whereas photons the particles of light are massless and therefore can travel at light speed c.
The LHC accelerates protons at 99.9999%…. the speed of light using up to 14Tev!! of energy but never succeeds at 100% the speed of light. And that is for the tiny tiny rest mass of protons, imagine how gargantuan much more energy would be needed the LHC to accelerate one molecule of your body at that speed? and then multiply that by the mass of the whole spacecraft to get an idea.
Achieving a speed say 1% of c then shutting down your engines will because the law of conservation of momentum in space, maintain you at 1% the speed of light but firing the engines back again does not mean that you will start to accelerate again and start increase with the same power say at 1.1% of c. Now you need even more power than before to accelerate to 1.1% of the speed of light and so on…
Currently the fastest man made object in space is Voyager 1 traveling at 17 Km/s constant speed that is 0.0056% the speed of light c. That is less than 6/100,000 or 60ppm of the speed of of light c=300,000 Km/s!!
A: The most close to correct is (2). 
To measure acceleration a pilot should just measure G-force (over weight). If he assumes some overweight as some acceleration and will sum it linearly, he will obtain more-than-c values at some time.
Moreover, a pilot will see, how surrounding Universe passing by faster than C. For example, if some mileposts are installed across Universe, he will see how they counting miles faster than C. 
This is because Universe is contracted along pilot's trajectory.
Another point of view says that pilot's time is slowing, but he doesn't notice this and hence feel that mileposts passing faster. 
Simultaneously, if the pilot will mark some star or planet and will measure the speed of this body, the result of measurement will say a value slower-than-c.
