First of all, kudos on a well designed game. It's obvious you've made something pretty addictive, as evident by the place on HN and the responses you're getting here.

One thing I've found out is that you can pretty easily get to at least 512 or higher just by repeating the following pattern:

  right + down + left + down

Try it out and you'll quickly see how it works. Any other similar pattern would also work:

  right + down + right + up
  up + right + up + left
  left + down + left + up

I will also sometimes break the pattern to consolidate some of the larger numbers when opportunities present themselves. But other than that, I usually stick to the pattern.

Of course, it will only get you so far, because you will eventually run out of space to keep the pattern working. But it will get you past the first few thousand points (512 or higher).