The Siege of Tyre has probably got to be close to top of the list for sure. It took substantial perseverance, patience and determination to accomplish what had never been done before - in itself something that Alexander strived towards.

One could also possibly say that to Alexander himself taking an army further East than Dionysus, Heracles or Achilles was his greatest achievement (and satisfying his insatiable 'pothos' at the same time).