New Elementary Function?

I think one of the candidates may be the error function. It is defined as

$$\operatorname{erf}(x)=\frac{2}{\sqrt\pi}\int_0^xe^{-t^2}\,dt$$

It is easy to understand as an integral of $e^{-x^2}$. The constant $2/\sqrt\pi$ comes from the fact that

$$\int_0^\infty e^{-x^2}\,dx=\frac{\sqrt\pi}{2}$$

and it forces the function to have limit at $\infty$ equal to $1$. Its derivative is elementary, namely

$$\operatorname{erf}'(x)=\frac{2e^{-x^2}}{\sqrt\pi}$$

and its integral may be expressed using elementary function and this function itself,

$$\int\operatorname{erf}(x)\,dx=x\operatorname{erf}(x)+\frac{e^{-x^2}}{\sqrt\pi}+C$$

It is used in probability and calculus.