Question: As I understand, an object with mass cannot reach the speed of light because its (relativistic) mass increases "exponentially" as it approaches light speed.

So there is a relation between mass and speed. But if speed is a relative measurement (it depends on a point of reference) how does that affect the mass?  

Is the mass also relative to a point of reference?  

Or is there some absolute speed (wrt some fixed point) which affects the mass of an object?

***Follow-up***:  

What I have trouble to understand is... given these two facts (I hope I got the facts right):

1. Light always moves at the same speed (independently of the inertial
frame of reference)
2. The speed of a mass object is relative to a frame of reference

then if an object is moving at 99.999% of light-speed for a given observer, what will a different observer see?  

Will that other observer see the object moving at a different percentage of light speed?  

(assuming the observers are moving at different speeds, of course)

Answer: 
The answer by Alfred Centauri is correct, it is how physicists now view the use of the term "mass", but it needs some clarification on the usage.

$$\vec p = \frac{m\vec v}{\sqrt{1 - \frac{v^2}{c^2}}} \tag{1}$$

When special relativity was first studied the equation 

$$E = \frac{m{c^2}}{\sqrt{1 - \frac{v^2}{c^2}}} \tag{2}$$

(where m is the rest mass the measure of the four dimensional vector describing the particle/system in motion), gave a new definition for the energy of moving objects.

From Newtonian physics we know that $\vec{p} = m \vec{v}$, so it was "natural" to think of the value multiplying the velocity vector in momentum as a mass: they called it "relativistic mass" because as far as classical inertial systems go the particle will behave to changes in motion as if its mass is increasing according to formula 2): more and more energy has to be supplied to reach the value of v. It is the relativistic mass that increases. In the rest mass of the particle, its mass is always m.

The use of the term "relativistic mass" is being dropped, because it gives rise to confusions as in your question. Each particle has a mass in its center of mass system that does not change . 

A better term might be "apparent mass", how an observer would see a particle approaching the speed of light : as if it had more and more difficulty reaching there.

By the way, in the lab we observe relativistic elementary particles which are point particles, like the electrons.

> 
> T[he particle accelerator known](http://www.penguin.com/ajax/books/excerpt/9780142001868) as the Large Electron Positron (LEP) collider at the Centre Européenne pour la Recherche Nucléaire (CERN) laboratory near Geneva could propel electrons to 99.999999999 percent of the speed of light.
> 
> 
> 

Their relativistic mass increases, but they are still measured as point particles with rest mass m.
If a baseball was passing right in front of you but suddenly all time surrounding you came almost to a standstill, you would see the ball hovering in the air in front of you. The ball is now in a very slow time frame. 

Due to it being in a slow time frame, if you try to push it or grab it and throw it, an enormous amount of energy would be required to do so. Even if you did manage to throw the ball at a typical speed from your point of view, from its point of view it is moving at an enormous speed since it is now crossing large distances in very short time periods, thus an enormous amount of energy would be required to throw the ball while at the same time be satisfying the laws of physics as seen from the balls point of view.

Thus the closer an object approaches the speed of light, the slower a time frame it is existing within. This therefore simulates there having been an increase in its mass.

How much of an increase of mass there seems to be, all depends upon how close your frame of reference is to that of the object, meaning how fast is your clock ticking relative to the clock of the object.

As mentioned by physicist Brian Greene in his book The Elegant Universe ( See The\_Elegant\_Universe-B.Greene.pdf Motion Through Space-Time pages 26 and 27 ) all objects are constantly on the move within Space-Time at the speed of light.

Thus it is to be noted that all objects are moving at the speed of light to begin with. What can be done however, is change the direction of that constant motion within the 4 dimensional Space-Time environment.