Lickorish and Martin 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
BingFurch's 1924 paper (discussed in Bing's survey cited by Lickorish and Martin)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 also constructed triangulations of the n-ball, n≥3$n$-ball, $n\ge 3$, whose rth $r$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 (see their Corollary 3.5).