Probably the most natural examples of reflexive spaces that are not super-reflexive are the spaces $(\bigoplus_{n=1}^\infty\ell_q^n)_{\ell_p}$, where $q\in\{1,\infty\}$ and $1<p<\infty$. They are reflexive as the $\ell_p$-direct sum of a sequence of finite dimensional spaces, but not super-reflexive since they contain isometric copies of $\ell_q^n$ for all $n$.

Probably the most natural examples of reflexive spaces that are not super-reflexive are the spaces $(\bigoplus_{n=1}^\infty\ell_q^n)_{\ell_p}$, where $q\in\{1,\infty\}$ and $1<p<\infty$. They are reflexive as the $\ell_p$-direct sum of a sequence of finite dimensional spaces, but not super-reflexive since they contain isometric copies of $\ell_q^n$ for all $n$.

Tsirelson's space is also an example of a reflexive space that is not super-reflexive, but obviously it is not quite so natural since it was invented to be a Banach space having no subspace isomorphic to $\ell_p$ or $c_0$. Figiel and Johnson later constructed a super-reflexive space having no subspace isomorphic to $\ell_p$ or $c_0$.