Depends on your definition of non-trivial and reasonably convex. After playing with the tiles a little bit it seems to me that placing 4 (maybe even 3) hats on

Depends on your definition of non-trivial and reasonably convex. After playing with the tiles a little bit it seems to me that placing 4 (maybe even 3) hats on top of each other is one such configuration. You can place 1-2 more tiles on each side, but you quickly endup with state where the dead end is obvious.