I've often heard that we can give examples of CW complexes that aren't homeomorphic to the realization of any simplicial set (although I haven't heard that there exist Kan complexes that aren't isomorphic to the total singular complex of a CGWH space. Are there?) Would someone mind providing an example of one (and an example for the opposite statement as well, if it is true)?

The mapping cylinder of a really messy continuous map $I\to I$ The nerve of the category in which there are two objects and each Hom set is a singleton. 


The geometric realization of a simplicial set is always triangulable. See Corollary 4.6.12 in Cellular Structures in Topology by Fritsch and Piccinini. They also give an explicit example (in section 3.4) of a nontriangulable CWcomplex (which uses, I think, essentially the same idea as Tom Goodwillie's suggestion). This paper contains another example and shows, on the other hand, that every CWcomplex with cells in at most two dimensions is triangulable. 

