When camping at the campsite at Vesperpool, Gladio will ask Noctis if he's up to fishing the Liege of the Lake, a big fish, no one ever caught before.
This is the setup I had:
Level 8 at fishing
Dragon's beard (Line)
Sweet Jamming Mousse (lure for Glowing Barrelfish)
Death Spin (Rod)
It took me a couple of campings to catch this one. It can happen that you fish the lake empty and have to come again.