After the events on Planet Styx, you'll end up in area called Gemity. This area is essentially a collection of mini-games, head to the main plaza and go either to the Northeast or Northwest transporter to access Runic Chess. If you forgot how to play Runic Chess from your time in the Ancient Ruins of Mosel, the rules are as follows:
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There are a total of 300 Battle Trophies in the game, to unlock this trophy you need to obtain 240 of all of BT's: that's a small margin for error considering a good amount of the BT's can only be obtained on Universe and 4D difficulty. That being said, it is possible to obtain all 240 BT's on Galaxy mode, though it will be difficult. You'll have to get all of the story related/boss trophies and keep in mind the missable BT's like the Albel Boss Fights and Norton Redux fight. This FAQ is a great resource to help you on the specifics of each BT, below I will detail some things to keep in mind while working on this trophy:
When it comes down to your main party, try to stick with only them for the duration of the game. Keeping characters as low leveled as possible helps later on for BT's where you need to level up consecutively 20 times or fight solo.Speaking of levels, in regards to the “Recover “X” amount of health” BT's make sure you keep Mirage and Roger at their initial level so that you can get these BT's easily. (Make spare saves if you need to)The 400 types of enemy BT may seem like a lot at first, but if you're constantly killing every enemy you encounter and return to those areas later in the game (the enemies in a lot of areas change after you finish the Styx area). you can rack up many of those BT's just by playing through the main game. Additionally, you can also score a few more types by playing through the Battle Arena. Assuming you kept skipping ahead every 4 ranks, you missed quite a few enemy types by not fighting some of the teams in Ranking Battle as well.
Luckily, a large majority of BT's are not missable, so once you get all the missable BT's out of the way you can save the rest of your grind for post game. Once you have obtained 240 Battle Trophies, the trophy will unlock. (Credit to Aerius for the BT Guide)