Insect Extermination - "There has been a major outbreak of Insects in the hills. Time to exterminate them"
Starship Anchor (Seven Wonder) - "Reminiscent of an anchor, it points toward a polar star. Legend has it that this was the anchor on the god's starfaring ship"
The Secret Dungeon can only be accessed from the Chapa Elder if all main quests and side-quests have been completed without finishing the game. This is mainly used by Armic in his quest to collect the ingredients for the Rain ceremony, but it can also be accessed by Ventus, Laura, Kurt and Ruby if the right conditions are met at specific points in the game. However, this does mean for the other characters that you will have to finish some tough side-quests (inlcuding some of The Seven Wonders) early in the game.
Both Judy and Mythe cannot access the bonus dungeon as they both never visit Chapa Village.