Tales of Vesperia

You first encounter a Giganto Monster in the Weasand of Cados if you are simply following the main storyline. It is called the Hermit Drill and if you are around Lv. 35-37 when you try the fight you should be able to take him out fairly easily be using the Blue Arrow Artes in Overdrive to make him dizzy and make sure to hit the Fatal Strikes when they show up. Use Lemon and Orange gels as neccessary to keep health and TP high. I played through the entire game without finding out about the King of Adventures title sidequest for giganto monsters and both achievements still popped anyways.

Kill your first Giganto Monster. Note that to get the quest you must not fight one until you get the quest from the roving inn.