We illuminate the boundary between decidability and undecidability for hybrid systems. Adding any of the following decorations to a timed automaton makes the reachability problem undecidable: 1. a single stopwatch with weak (less than or equal to, greater than or equal to) edge guards 2. a single skewed clock with variable equality tests 3. a single two-slope clock with weak edge guards 4. a single memory cell with weak edge guards As a corollary, we obtain undecidability for linear hybrid systems with triangular differential inclusions, which have invariants of the form x' less than or equal to y'.