Non-trivial values of error function $\operatorname{erf}(x)$? The so called error function $\operatorname{erf}(x)$ is defined as
$$\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}dt,$$
and it is well known that $\operatorname{erf}(\infty)=1$.

Are there any other known closed-form values of $\operatorname{erf}(x)$, except for $\operatorname{erf}(0)$ and $\operatorname{erf}(\pm\infty)$?
If the values aren't listed on the Wolfram function page, I would be surprised if you found them anywhere else. The only listed closed form values are for $0$, $\pm\infty$, and $\pm i\infty$. However, you can find various equivalent formulations, continued fractions, and the like on that page. A good reference, generally, for most well-known functions.
There are other exact values of erf(x) besides 0 and plus or minus infinity
this graph there are horizontal asymptotes at y=1 and y=-1 because



The Z-score is a probability that certain data fall within a certain interval. For example a Z-score of 1 is represented by erf(1) = 0.682 which means that plus or minus 1 standard deviation from the mean will encompass approximately 68.2% of the population based upon a sample population. A Z-score of 2 erf(2) = 0.955, erf(3)= 0.9995. In order to be sure that you have covered the entire population or be 100% certain about your data you have to cover an infinite number of standard deviations, which is why it would be impossible to do such a thing.

Example mean IQ is 100 and SD is 15, since erf(3) = 0.9995 then 99.95% of the population will have an IQ within 3 standard deviations from the mean.



STANDARD NORMAL DISTRIBUTION: Table Values Represent AREA to the LEFT of the Z score. https://www.stat.tamu.edu/~lzhou/stat302/standardnormaltable