Freedman's $E_8$-manifold is nontriangulable, as proved on page (xvi) of the Akbulut-McCarthy 1990 Princeton Mathematical Notes "Casson's invariant for oriented homology 3-spheres". 
Kirby showed that a compact 4-manifold has a handlebody structure if and only if it is smoothable:
[1] and [2]. When is a compact topological 4-manifold a CW complex? 


  [1]: http://www.manifoldatlas.him.uni-bonn.de/index.php/Questions_about_surgery_theory
  [2]: http://mathoverflow.net/questions/7921/failure-of-smoothing-theory-for-topological-4-manifolds/38739#38739