Why don't we use rapidity instead of velocity? In school we learn that we can add velocities together, and then later on we learn that it's not correct and that there is a speed limit. Why create all this confusion when we could just use rapidity to begin with?
Rapidity is defined as $w = \mathrm{arctanh}(v / c)$, where $v$ is velocity and $c$ is the speed of light in a vacuum. Rapidities can be summed and have no upper bound. At non-relativistic speeds it acts proportional to velocity.
In fact, at non-relativistic speeds, we could substitute $v$ for $wc$ (rapidity times speed of light), and one could hardly tell the difference. The ISS moves rather fast at a velocity of 7660 m/s (27,576 km/h), and has a $wc$ of about 7660.0000016667 m/s. Why can't we just substitute velocity for rapidity in real-world and classroom use, and end the confusion about why there is a speed limit once and for all?
 A: My guess would be that people like to use the simplest tools for doing specific tasks, until it turns out that there is a special task for which they require a more sophisticated tool. Non-relativistic speeds do behave as if they were very simple vector quantities, so why go through the rapidity overhead just to get a negligible improvement of your results? 
Pretty much everything we believe and calculate in the real world is based on approximations (e.g. small angle approximation, Taylor series, the derivative) and numerical solutions to differential equations (Fourier series, finite element simulations), so the takeaway would be to simplify where possible and to complicate only where it's absolutely necessary.
A: Whatever you choose as you everyday quantity, you'll eventually have to use velocity. And that's needed in quite a natural context: to predict how much time it would take for an object you observe to cover some distance.
An example of a practical question: What is the minimum time for a space probe to reach a planet in Alpha Centauri and return back with a sample of its surface? Rapidity is useless here, and starting from it would needlessly add complexity to the calculations, which with velocity amount to division of double distance by mean speed.
On the contrary, adding velocities of two objects that both move at relativistic speeds is far from being a common practical problem, so exchanging the ease of use of velocity in everyday problems for (seeming) intuitiveness of rapidity doesn't really look useful.
A: In everyday life, we experience the universe in a non-relativistic classical way. 
We are familiar with the concept of time and space. 
Defining the velocity as the ratio between a distance traveled in a given time interval is a much more natural choice instead of defining the rapidity.
If we all lived at relativistic speeds, or close to the event horizon of a black hole, or if we were small as an atom, we would use other tools to describe the universe around us. 
However, in our case, the quantities of classical mechanics work quite well and we can have a direct intuitive grasp of their meaning.
A: Speed of light isn't a limit on abstract velocity, only on the velocity attainable by a body. $2c$ is a mathematically valid velocity. A spot of light created by a laser pointer shining upon a surface can move faster than the speed of light. Velocities are in fact perfectly summable.
If I observe some object moving $0.8c$, I can think about adding another $0.8c$ to make it go at $1.6c$. That will turn out to be energetically impossible, but the velocity itself is cheerfully ponderable and expressible.
A: It is not particularly easy to add rapidities pointing in different directions. 
e.g. Suppose B moves with rapidity $\rho_{1}$ with respect to A in the common $x$-direction of both of them. And suppose that C moves relative to B with rapidity $\rho_{2}$ in the common $y$-direction of B and C (we are taking it that their axes are aligned in this way). Now what is the rapidity of C relative to A?
You will not find it easy to answer.
The fact that this sort of thing is non-trivial is another reason (in addition to some good ones given in other answers) why it does not help to replace velocity with rapidity.
A: 
People understand what velocity means, and how it is used.  Anyone who needs to use relativistic models knows enough to distinguish between which model they are using.  No additional words were needed when it was created,  and trying to invent new words now would just cause more confusion, not less.
A: While this varies by country, it's fair to say that kids have a qualitative understanding of speed before kindergarten.
By grade 8, they're talking about speed, acceleration, and related issues of energy quantitatively.
Vectors are introduced during high school.
Most students won't get a quantitative discussion of Special Relativity until freshman physics.
Hyperbolic trig functions won't appear until AP Calculus or freshman year as well.
Therefore, introducing rapidity in grade 8 is only really going to be feasible in a non-relativistic way. As you stated in your question, one could present rapidity in its product form with c. However, all you've achieved here is a change of terminology; students are still using velocity in its non-relativistic form.
We're already teaching them about velocity in its non-relativistic form. We call it velocity.
What seems intuitive, in retrospect, is a lot less intuitive as you're learning it. Most physics students will never be exposed to SR. Those that are should be advanced enough to understand that SR corresponds to classical mechanics at low speeds, in the same way that they will understand that quantum mechanics corresponds to classical mechanics at high energies. This understanding is an essential step in their growth as physicists.
A: As well as issues of practicality, it doesn't answer the question of why there's a speed limit. (It can't as it's just a mathematical transformation.).  The question becomes 'in the formula for $w$, why do you take $c=3 \times 10^8$ m/s'?
A: In retrospect, having seen special relativity, what you suggest is reasonable. In fact, one can unify the geometries of Euclid, Minkowski, and Galilean-relativity after one distinguishes that "Galilean rapidity" is different from "[Minkowskian] rapidity".
So, I think we should use at least introduce it (or aspects of it) if one is going to discuss special relativity... which is one way to reveal that we are reaching the limits of an approximation. (Why not use GM/R^2 instead of g?)
Here are some reasons why we don't use rapidity.


*

*Historically, we didn't [and primarily still do not] view a position-vs-time graph as a "[Galilean] spacetime diagram" and thus make connections to Euclidean geometry.

*Rapidity is an angle (a spacelike arc-"length" [with the appropriate metric] or a sector area in an appropriate "circle"), whereas velocity is a slope... and we tend to think of "rates of change".

*One could argue that velocity is "more physical" (in our non-relativistic upbringing) than rapidity which is arguably too abstract. (Edwin Taylor told me that he dropped rapidity from the 2nd edition of Spacetime Physics because its users (teachers) reported to him that they didn't use it. A few of us politely protested and suggested that he put it back in a future edition.)

*Some could argue that trigonometry is hard... especially hyperbolic trigonometry.

*There are likely others... but I keep some of these in mind as I develop an approach to using a unified viewpoint suggested above to teaching physics, dropping such hints along the way... with the goal of making relativity more understandable, and less mysterious.


It's probably not too far off to suggest that the following is an analogue of your suggestion.


*

*Why not use aspects of spherical trigonometry (e.g. angles on the sphere to measure distances) instead of Euclidean geometry of the plane?


UPDATE:
In trying to motivate rapidity (in relativity) vs velocity (in relativity)


*

*Rapidities add but velocities don't... just like saying that angles add, but slopes don't: the slope between two lines is not the difference of their slopes. (To rotate images, does Photoshop use slopes or angles?)

*A lot of these cryptic calculations (e.g. Momentum in center of mass-frame out of knowledge kinetic energy in lab-frame ) are easier to interpret with rapidity... and appealing to one's intuition with ordinary trigonometry and Euclidean geometry.

A: I think the main reason, as already captured in part by other answers (e.g. Davide Dal Bosco's), is the following: velocity is a physical quantity, it tells us how far something goes in a given time.
Rapidity may be mathematically convenient due to its relativistic addition properties, but what does it tell us?
As an example, the rapidity of light is $w = \textrm{arctanh}(1) = \infty$. Isn't it much more useful to know that light moves at $c=299 792 458 \frac{m}{s}$ through space?
Mathematically, we can transform everything back and forth as we wish to simplify our calculations. But in the end, we will want to know something physical: the velocity.
A: There are really 2 questions here, so let's answer them separately.

Why don't we use rapidity in daily life?

Simple, most people don't even know what relativity is don't understand its implications or how to reason about them. They are not capable of using relativistic quantities.
If you are suggesting they simply don't worry about relativity and just be mindful of the $arctan$, the answer doesn't change. Most people do not understand trigonometry and cannot use it; including many people who have a need to understand and manipulate velocities. For example, how many people would still have a driver's license if this was required to pass driving exams?
If you are saying they should just call it "rapidity" and then act exactly as if they were speaking about non-relativistic velocities, then sure that could work. But it would get in their way if they do start learning advanced physics and must differentiate between the "old rapidity" and the new one. Also, "rapidity" sounds kind of funny.

Why don't we teach rapidity in school?

Most students who learn about velocities do not go on to learn relativity, so they would never get the pay off. On the other hand, those that do learn relativity, probably do not stop at scratching the surface of its definition. They would go on to learn more advanced topics. If we are asking these students to comprehend those advanced topics, I think the simple matter of velocity in a relativistic context is not that confusing and arguably not worth the extra terminology.
A: There are several answers to this question.
Rapidity is not taught from the outset in introductory physics classes in part because it would unnecessarily confuse students, and besides, you only need to worry about it when you are dealing with relativistic speeds (or precise-enough measurements that can detect relativistic effects). But there are other reasons, too.
Even when working in SR (special relativity), rapidity is not as useful or (in some sense) as fundamental as velocity. It's true that, in SR, for relative motion in 1D, one simply adds rapidities. But what if you want to know the position of an object after a certain amount of time has elapsed (given its initial position)? For that, you'll need a velocity.
Even then, in most cases in which rapidity would come in handy, it is better to deal with momentum (or 4-momentum), since this is what is actually conserved. This is connected with rapidity very simply: the 3-momentum is a vector pointing in the direction of motion, with a magnitude equal to the rest mass times the rapidity (and the time-component of the 4-momentum is the total relativistic energy).
Also, generally, practicing physics is the art of keeping things as simple as possible. Life is complicated enough. There is no reason to introduce relativity, quantum mechanics, warped spacetime, etc etc, if you don't really need to.
Finally, historically, Einstein's special relativity, Lorentz transforms, Minkowski spacetime, etc, were developed long after Galilean relativity and Newtonian mechanics. In most physics courses it makes sense to pursue a pedagogical track that more-or-less follows the historical development because then students can see how the accumulation of evidence and lines of reasoning led, historically, to improvements on what came before. Physics, like all sciences, does not arise from pure reasoning from a set of postulates handed down on stone tablets. It is important that students understand that the theory of SR arose out a failure of classical physics (Newton + Maxwell) that led to contradictions that needed to be resolved.
A: Why does the US uses imperial system and not metric system?It's just that the population understands what a mile is better than a kilometre because people have used it since birth. NASA and most scientific community still use metric system.If you know what metric equivalent is then surely you can convert it.Its better to use formulas to convert than to ask whole population to change to metric. 
Similarly,most of people can understand what velocity is.Someone who can understand the difference between velocity and rapidity can learn to convert from one to other easily.
Surely, it would be easier than asking a whole generation of people to understand rapidity, its difference from velocity and why does not have a upper limit?
