It is difficult to take just one 'deed' and say this is the greatest. There are many acts of heroism in Tolkien's books so here's one Ithink is an act of heroism, maybe not the greatest, but heroic nonetheless.
At the Council of Elrond Frodo steps forth and says he will take the Ring to Mordor. He knows little of the evils of Sauron, knows little about the other races, knows almost nothing about the rest of Middle-earth yet he (maybe in ignorance) says he will do this thing. From then on his life focuses on one thing - to get the Ring to Mount Doom and destroy it. He is no warrior, he is a hobbit. He has no training in fighting, yet he volunteers to do this. Other than BIlbo, he was the only one to volunteer.
I think that was pretty heroic.