Let $\operatorname{sn}$, $\operatorname{cn}$, $\operatorname{dn}$ be Jacbian elliptic functions (https://dlmf.nist.gov/22). According to Greenhill,
The integrals $$\operatorname{sn}^2u,\operatorname{cn}^2u,\operatorname{dn}^2u$$ require for their expression new functions called elliptic integrals of the second kind, such as occur for instance in the rectification of the ellipse.
Can we prove that $$\int \operatorname{sn}^2u\,\mathrm du,\ldots$$ cannot be expressed as a finite composition of elementary functions and elliptic functions? This is equivalent to proving that the Jacobi epsilon (https://dlmf.nist.gov/22.16#ii) cannot be expressed as a finite composition of elementary and elliptic functions.
You can contrast this with, e.g., $$\int \operatorname{sn}(u,k)\,\mathrm du=-\frac{1}{k}\operatorname{artanh}(k\operatorname{cd}(u,k))+C$$ where no elliptic integrals of the second kind appear and where elementary functions and elliptic functions are sufficient.