[Lickorish and Martin][1] constructed, for each $r$, a triangulation of the $3$-ball whose $r$th barycentric subdivision collapses, but $(r-1)$th doesn't. The basic idea, going back to [Furch][2] and Bing, is to triangulate a cube with a knotted hole, where the missing knot has bridge index $2^r+1$, and then fill back a small part of the hole - so that topologically, no hole remains, but the $3$-ball now contains a knot triangulated by a single edge.

**Added later:** [Kearton and Lickorish][3] also constructed triangulations of the $n$-ball, $n\ge 3$, whose $r$th barycentric subdivision is not collapsible. On the other hand, every triangulation of a ball becomes collapsible after some number of barycentric subdivisions, according to a recent preprint by [Adiprassito and Benedetti][4] (see their Corollary 3.5).


  [1]: http://www.ams.org/journals/tran/1969-137-00/S0002-9947-1969-0238288-X/S0002-9947-1969-0238288-X.pdf
  [2]: http://infoshako.sk.tsukuba.ac.jp/~HACHI/math/library/knot_eng.html
  [3]: http://www.ams.org/journals/tran/1972-170-00/S0002-9947-1972-0310899-2/
  [4]: http://arxiv.org/abs/1202.6606