A few additions to Marcin's writeup.
1. Sometimes, at very end of each level when you had to jump to the flagpole in order to advance to the next level, it was possible to actully jump over the flagpole. It was just a matter of über-accurate timing and adequate speed at the critical moment. You could then run ahead ad infinitum. As the game just wouldn't scroll backwards, you had to start a new game if the flagpole had already disappeared to the left. If it was still visible, you could still finish the level simply by jumping to it from The Other Side.
2. Really an addition to bug #6, from the list above. Little fireball-spitting Mario could be achieved at the end of castle levels, by simultaniously touching the Boss and the axe-looking thingy at the far end of the bridge. In order for this to work, you must alrady have eaten the funky mushroom to grow big, or else you'll simply die (I think. It has been a while since I last tried this).