When it comes to complex analytic functions on open subsets of $\mathbb{C}$, it is hard to come up with examples of pointwise convergent sequences that do not converge uniformly on compact sets. That is partly because it doesn't take much for a family of analytic functions to be normal. There is much more to be said about the matter, and for an exposition including examples I recommend this survey by KrantzDavidson.(If you're really interested, you'll want to look at the first 2 references listed there for more information.)
[Edit: This post would be better if it included a description of such an example. I may add one when I have more time.]
