Is there anything "special" about elementary functions? I just found an article on Liouville's integrability criterion, which gave me a thought.
What makes functions like $\mathrm{Si}(x)$, $\mathrm{Ei}(x)$, $\mathrm{erfc}(x)$, etc. inherently different from $\sin{x}$, $\log{x}$, etc. ? 
Related question: What is the exact definition for the elementary field, and what is its significance to this question?

Really it is somewhat arbitrary.  We don't usually include things like Si(x) etc. because they are used only in special situations.  Whereas $e^x$ is everywhere -- no matter where you go you cannot escape it.
There are many, many, many special functions.  So many that it is possible to take an entire course in special functions (which I once did) -- and that no doubt left out most of them.
Still, at least one eminent mathematician hated special functions and told me this story about why:
A lady was given a tour of an observatory, and the tour guide started explaining about many of the stars she could see through the telescope.  When the tour was over, the lady thanked the guide and said "I'm really impressed with the work you do here.  And what impresses me most of all is that you know the names of all those stars".
Well that is one opinion, and I think something of an outlier.  I like special functions because they allow us to write complex things with simple notation. 
