An often-cited example of a very basic program, for which termination is open, is one that generates the [Collatz sequence][1].  In pseudo-code:

<pre><code>collatz(int n)
  while (n != 0)
    if (n mod 2 == 0)
      n = n / 2;
    else
      n = 3n + 1;
</code></pre>

  [1]: https://en.wikipedia.org/wiki/Collatz_conjecture "Collatz sequence"