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?

Yes, beginning from constants and the identity function, the different elementary functions are obtained by applying closure with respect to different elementary operations. Close by sum and multiplication and we get polynomials. Close by solving linear differential equations or order one, with constant coefficients, and we get the trigonometric and exponentials. Close by computing inverse function and we get logarithm and arc-trigonometric. We get those you mention after closing by integration.
