One of the more important parts of the game, especially in the late endgame, is the need to power up your equipment. Note that this page will use the Carnage equivalent of the Trapezohedron as an example, since that is what most people who are doing this will use as their main piece of equipment. This is because it actually has stats for all of the attributes and is relatively the easiest piece of equipment to max out. Before starting, you need to procure one, which can be done by stealing it from the Item God 2 of the Carnage Arcadia.
However, finding an Arcadia can be a bit challenging, as there is no surefire way to get one without having to rely on some randomness. One way to do this is by diving into a high ranking item, either a Rank 38 or 39 (giving yourself a greater chance of one appearing). Of course, this might be a little challenging, as you're going to need an item of that rank. Luckily, you can manipulate things a bit, with the Eryngi Fluid Evility from the Shroom class. Go to a Carnage stage where you can take some damage (misses or 0 damage doesn't proc it) and keep going until you get a Carnage Mushroom Soup, which is Rank 39.
Now that you have that out of the way and have gotten the Carnage Arcadia, go into it and level it up until you can trigger the appearance of the Item God 2 (level 80+). Hopefully you've done the necessary building of a character (40 million stats), as you will need it to steal the Trapezohedron. Don't forget that you can buff the stealing character's HIT and debuff the IG2's SPD to give yourself a greater chance at actually stealing the Trap. With the Trap in your possession, it's time to start increasing its strength.
Leveling the Trapezohedron
The first thing that should be said is that you cannot screw up leveling an item in Disgaea 5, unlike previous games in the series, since floors are endless and you can keep killing enemies. There are two important parts to leveling the Trap, the Kill Bonus and the Training Bonus. The former is gotten by simply killing things, with it increasing more as the enemy's stats are higher. The latter is gotten by killing all Invaders (should they appear), clearing any battle in a Mystery Room, and killing Item Bosses/Proto Darkdeath.
The Sage is the best class in clearing the Item World, as Land Decimator will hit all enemies and if it's strong enough/with enough support, you should be able to clear most (if not all) of the enemies on any given floor. Most of what you need to do is already outlined in the What's Next? section; you want the Sage with 40 million stats, as well as the four support characters talked about there, as you will definitely need them.
Note: It's a great idea to have a back-up character who has been trained as a single-target killer, such as a Magic Knight with anything that is designed to augment that single-target damage, like Bushido and Concentration.
The next part is up to you and what you can handle, as everyone seems to take things differently. The ideal setup would be to set the Enemy Strength in the Cheat Shop to 20 stars and go to town on the enemies, as this would optimize the Kill Bonus. Of course, if you can't deal with that, you can set the stars to a lower value. With Explorer on your four supporting characters and as long as they are out whenever you clear/level a floor, you will get Mystery Rooms, which will help pump up the Training Bonus (something to focus on later on, but anything helps for now).
To give you an idea of how to go about clearing each stage, you will bring out your support characters to surround the Base Panel, then bring out your Sage and buff her with Magic Boost and Multi-Attacker, then without moving on the Sage, use Land Decimator. If the enemies are not killed, then you can use Afternoon Tea on the Sage and do Land Decimator once more. You will be doing this for a good while, with the main goal of getting the current item to level 500.
While you can extend the level past 500 (only applicable if you've cleared the Carnage Quest Line) via the Item Assembly, it's not needed, as you can continue diving into the Trap to continue leveling its stats. Keep going until you get the Kill Bonus to 400, then you can concentrate on the Training Bonus. The catch about the Training Bonus is that it is maxed out for a particular stat when you can no longer pass the corresponding bill for that stat in the Item Assembly. Once you have the Kill Bonus at 400, you can set the Enemy Strength back down to zero stars.
The max stats that Trap can get is around 2 million for each stat and about 400 million for HP. The best thing to do is to not worry about the individual stats and concentrate on passing bills for the HP, but Movement and Jump are also two good additions to pass. With the Enemy Strength back to zero, you can simply skip the floors and concentrate on the Mystery Rooms and Item Bosses to get your Training Bonus up. One quick way is to set the Item Boss option in the Cheat Shop to ON and use Diver-10s to skip straight to the Item Bosses. Once you max out both the Kill and Training Bonuses, then you can go hunting for the Duplication Mystery Room to try and dupe the now maxed Trap.