Why can't we define more elementary functions? $\newcommand{\lax}{\operatorname{lax}}$
Liouville's theorem is well known and it asserts that: 

The antiderivatives of certain elementary functions cannot themselves be expressed as elementary functions.

The problem I got from this is what is an elementary function? Who defines them? How do we define them?
Someone can, for example, say that there is a function which is called $\lax(\cdot)$ which is defined as:
$$
\lax\left(x\right)=\int_{0}^{x}\exp(-t^2)\mathrm{d}t.
$$
Then, we can say that $\lax(\cdot)$ is a new elementary function much like $\exp(\cdot)$ and $\log(\cdot)$, $\cdots$.
I just do not get elementary functions and what the reasons are to define certain functions as elementary.
Maybe I should read some papers or books before posting this question. Should I? I just would like to get some help from you. 

The reason for the definition is because when you study properties of a class of functions, then it is very convenient if this class of functions has a name. Liouville used the name "elementary functions" because at the time, that name was still available.
You can certainly define a different class of functions. Your only problem is that the name "elementary functions" has already been taken, so you'll have to use a different name for your class of functions.
