A simple and natural example of such a space is that of the row-finite matrices. These play a central row in the theory of summation of divergent series since many of the classical methods are implemented by such matrices (prominent example---Cesaro summation and its higher degree variants). The above space has a natural locally convex structure and it enjoys the requested property.
EDIT after Tom's comment. Am typing on my iPad and so can't use many math symbols. The standard reference on infinite matrices and summability is "Infinite matrices and sequence spaces" by R. Cooke (now a bit dated). On nuclear spaces, spaces of operators and tensor products it is Grothendieck's thesis. If you are interested in such topics, I recommend its study, a potentially life-changing experience.
The basic spaces involved are the nuclear Fréchet space $\omega$ of all sequences and its dual $\phi$, the nuclear Silva space of all finite sequences. The row finite matrices map the former into itself and it is easy to see that all such continuous linear mappings arise in this way. The rest is just formal manipulation of tensor products and the corresponding operator spaces, one further ingredient being the fact that $\omega$ is isomorphic to $\omega \otimes \omega$ (but, notabene, not in any natural way).
Now if $E$ is a nuclear Fréchet space which is isomorphic to $E\times E$ and there are many such and we put $V=E\otimes E'$, then
$$L(V,V)=V'\otimes V=(E\otimes E')\otimes(E\otimes E')'=(E\otimes E')\otimes(E'\otimes E'')=(E\otimes E')\otimes(E'\otimes E)=(E\otimes E) \otimes(E'\otimes E')=E\otimes E'=V$$
(equality denote "is isomorphic to", $\otimes$ any tensor product since everything in sight is nuclear).