If $A$ is uncountable then Problem J(b), p. 240 of Kelley says that $\mathbf R^A$ (or $\mathbf C^A$) with the product topology (a.k.a. uniform convergence on finite subsets) is not compactly generated. This topology is locally convex, being generated by the seminorms $\|z\|_F=\sup_{a\in F}|z_a|$ for all finite $F\subset A$.

**Edit:** Also, if I'm not mistaken, Frölicher & Roulin's *Topologies faibles et topologies à génération compacte* shows that an infinite-dimensional separable Hilbert space with its weak topology is never compactly generated; and its Math Review adds that this remains true of the dual of any infinite-dimensional Banach space, with the weak* topology.