Great topic! As I look at people's responses I am pleasantly reminded of so many great parts of Tolkien's work. Most of the instances cited are where the hero is driven by desperation, despair, or some great need.

While they may not rate in the top five, there are some heroic acts that are done purely out of compassion, with an acknowlegement that there is no obvious need for the deed: heroic acts of staying the hand, rather than striking. I am thinking of both Frodo and Bilbo sparing Gollum; Frodo stopping the hobbits from killing Sharky.

Another act that can be considered heroic, especially for the character, is Gollum helping the hobbits at all & not fighting them to the death, or at the first chance throwing away his promise in order to murder & get the ring.