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.
I don't know many equations so I think I'll give an example. Suppose you wanted to travel to the future. The best way to do so is to travel close to the speed of light. So imagine that you gear up your rocket to such a speed that for you one year in the moving rocket is equal to a million years on earth. Now from the perspective of those on earth, you need to pack in more fuel to compensate for the million minus one extra years you will be travelling. However from both perspectives (those on earth and yours) the rocket is moving at the same high speed with respect to the earth. So from your perspective you are apparently carrying enough fuel to last for a million years although your journey lasts but one year. So perhaps that's because the mass of your rocket increases due to its speed, in your perspective. In that case what fuel was used to compensate in the increase in time from the stationary (not really stationary as we mentioned the earth) point of view is used in your moving point of view to compensate for increase in mass. So I guess the mass of moving objects do increase but only from the perspective of the one moving. However this entire answer is only conjecture on my part!!