I am looking for examples on double dual spaces. As I know $\ell_p $ is the double dual of $\ell_p$ for $1<p,q<\infty$, $\mathcal L_p $ is the double dual for $\mathcal L_p$ for $1<p,q<\infty$ and $\ell_\infty$ is the double dual of $c_0$, where $c_0$ is the space of sequences of numbers that converge to 0.
I would like to know other examples of the double dual spaces, especially of non-reflexive spaces.
There is a nice duality between $C(K)$ and $L_1(\mu)$-spaces. The double dual of $C(K)$ is of the form $C(L)$ for some huge compact space $L$. (Actually it is also isometric to $L_\infty(\nu)$ for some huge measure $\nu$.) The second dual of $L_1(\mu)$ is also of the form $L_1(\nu)$. However people rarely think of duals/biduals of these spaces like that. If $K$ is a scattered compact space, then $C(K)^*$ is nice: it is $\ell_1(K)$, so that $C(K)^{**} = \ell_\infty(K)$. This property characterises scattered compact spaces.
Another good example of a bidual space is $\mathscr{K}(\ell_p)^{**} = \mathscr{B}(\ell_p)$ for $p\in (1,\infty)$. In other words, bounded operators on $\ell_p$ are the second dual of the space compact operators on $\ell_p$. This extends to some more general Banach spaces.
The second dual can add very little; all quasi-reflexive spaces are good examples. Historically the first example is due to James (the James space). It is a certain sequence space whose dual is also a sequence space; actually $J^{**} = \mbox{span}\{J, \mathbf{1}\}$, where $\mathbf{1}$ is the constant sequence.