My favorite example is the halting problem for Conway's "FRACTRAN" programming language: given a finite sequence of fractions q1, q2, ...., q_n, does the procedure "starting with a given integer and keep successively multiplying by the first element in the sequence which results in the product still being an integer until none of them do" halt? In fact there is specific sequence of fractions that is quite short which can be interpreted as a Universal machine.