TITLE: What kind of functions cannot be described by the Taylor series? Why is this?
QUESTION [25 upvotes]: It's true that I'm not familiar with too many exotic functions, but I don't understand why there exist functions that cannot be described by a Taylor series?  What makes it okay to describe any particular functions with such a series? Is there any difference for different number sets? In the case of complex numbers maybe? 
Could somebody provide an example?

REPLY [4 votes]: In addition to all the comments here, I would like to add the curious Weierstrass function, which is known for its quality of being nowhere differentiable despite the fact that it is continuous everywhere:
$$ W(x) = \sum_{n=0}^\infty a^n\cos(b^n\pi x)$$
Consequently, it does not have a Taylor series.
You can find a visualization of $W$ here.