The modified Bessel function of the 1st kind $I_0$ is defined by $$ I_0(z)=\frac1\pi\int_0^{2\pi}e^{z\cos\theta}\,d\theta $$ and arises, among other places, in the probability density function of a noncentral $\chi^2(2)$ random variable.

The error function is defined by $$ \text{erf}(z)=\frac2{\sqrt\pi}\int_0^z e^{-t^2}dt $$ and is closely related to the cumulative distribution function for the standard normal distribution.

I believe these are not elementary functions, but

is either $I_0$ or $\text{erf}$ elementary "relative to" the other?

That is, if we were to add one of them to our repertoire of "elementary" functions, would the other one become "elementary"?