Taking down Lao Shan Lung for the first time. You take down a building-sized dragon with what is essentially a toothpick to it.
Then you find out, that's it's a baby, and the adults will most likely never be seen......
One for one of the monsters: in the Akantor ecology, the eponymous Akantor duels a Gravios. The Akantor not only resists a full blast of the Gravios' Heat Beam, it takes out the Gravios in one blow and then tosses the corpse like a rag doll. Then it roars triumphantly over the Gravios as the volcanoes in the background erupt. Seen here.
In Tri, driving off The Ceadeus. Even if you had help in the form of ancient weaponry, you still took down a sea-dragon larger than your base camp. You broke the mighty horns that caused earthquakes every time he rammed the ocean. You dodged torrents of water that would have ended your life not too long ago, and you made a dragon, who is deserving of the name "sea god", one that the Guild would need to send an army after, run away. And in doing so, you save Moga Village. In-story, the final cutscene marks this as your crowning achievement. A masterwork of bravery and skill that will be Spoken Of Forever. Now imagine doing this without using the ancient weapons, using only the armor you have on your back and the weapon you had spent hours forging just for moments like these.
Hunters can get mini-size Awesome moments whenever they can kill off the monsters either really quickly OR with a weapon that most people would advise you NOT to use.
Generally, killing any elder dragon counts as this. You, an ordinary human with no magical powers, no Supernatural Martial Arts, no artefact to do all the hard work for you, face down a creature that will at least fill the average room entirely, with access to nigh-supernatural elemental power, hide impenetratable to any normal weapon and the sheer strength to snap your whole body in two with a flick of its wrist. You go at this beast with nothing but good physical condition, weapons and armor you carved the ingredients for from the still-warm corpses of weaker monsters, whatever ragtag Band of Brothers you can pull together and sheer force of will. Kill it, and were this at all like real life, you would likely go down forever in history and legend. It's that time when you realize you'be come a long way from farming guts and beating off raptors.
A meta example really, but recently David Gibson said that Monster Hunter Tri G wouldn't get localized in the U.S. or Europe. Christian Svensson of Capcom said, "I'd like to see him cite a source."