One thing I have always liked about Persona 3 is its boss battles. Because there is a boss guarding each major checkpoint in both Tartarus and the Abyss of Time, as well as a number of story battles, there are quite a few of these in the game. The quality of these battles can vary greatly (some are fun, others boring), as does the difficulty (some boss battles are a cakewalk, but others are unfairly punishing), so there are some issues with consistency, but as a whole these battles add a lot of challenge and excitement to the game. They take an incredibly solid core combat system and push it to its limit, making them far more tactically engaging and fun than typical fare.
However, any praise I may give the boss battles of Persona 3 doesn't apply to the various Full Moon bosses and other story battles. For the most part, these battles are far too easy, since if you have reached the target floor in Tartarus that month, then the Full Moon boss is too weak for your current ability level (as a rule, the bosses you need to clear in order to make that target are stronger than that month's plot boss). Also, these bosses tend to overly rely on unusual gimmicks that eat up their turns, so they are usually only challenging if you accidentally stumble into the "bad" way to fight them, and are otherwise poorly designed. For example, one boss is split into two parts, one of which can revive the other, but while they are combined any damage is split between then evenly. As a result, by the time one of them is weak enough for the revival ability to matter, the one with the revival power is just as doomed as the other. In addition, there is no particular advantage for them when they are combined, so their entire gimmick of combining and separating is just a liability that adds little to the fight. Some other boss battles, such as those against enemy Persona-users, simply lack either complication or challenge, even though they are some of the most important to the plot. As a whole, many of these battles are disappointing.
If the story battles are disappointing because of an over-reliance on gimmicks and low difficulty, the battles in Tartarus may swing too far in the other direction. These battles are just straightforward battles against normal-looking enemies who don't make use of anything but normal attacks and ordinary elemental immunities, but they make use of these attacks and immunities in amazingly creative ways in order to provide difficult challenges. Unlike normal enemies and story bosses, you can't analyze a Tartarus boss's strengths and weaknesses, so they need to be determined through experimentation, adding a sometimes frustrating but often fun part of the battle (something resembling the Megaman "which weapon works?" game). In addition, these enemies play with the idea of what it means for something to be a "weakness". Rarely, they just have a normal weakness, which lets you abuse the "One More" and "All-Out Attack" abilities to defeat them quickly. Other times, they don't have a weakness at all, and you need to rely on critical hits in order to get All-Out Attacks. At other times, an enemy has so many resistances that the one thing they are not resistant to can be exploited as a "weakness". In some battles, the enemy has a weakness, but their attack power is so unfairly high that your only choice is to knock them all down and forego an All-Out Attack in order to leave them on the ground, slowly eating away at their health while keeping healed. A few bosses protect weaknesses behind active defensive spells like Tetrakarn and Makarakarn, which reflect physical or magical damage. At least one boss bends the idea of "weakness" so far that it actually takes almost no damage at all from the element it is weak to, so you are forced to make a decision between dealing damage and knocking it down. Many more rely on eating up player actions through status effects. Most of these tricks are used by normal enemies, so these boss battles are really nothing more than scaled up normal fights, but the raw offensive power of these bosses, which can destroy even a good team in only a few turns, combined with their large hitpoint totals, forces the player to carefully balance offense and defense in an efficient strategy in order to even survive, let alone win. Unlike far too many other games I have played, these battles really reward (or even require) good tactics and strategy, and their high difficulty adds a lot of excitement.
Of course, one issue about these boss battles is that they can sometimes be a little too brutal. Bosses that have no weaknesses and constantly attack with "Megido" spells (powerful attacks that hit all characters and have no element, meaning there is no defense against them) are unbelievably frustrating and far less fun than others. I suppose this is because there is literally no other way to beat such enemies other than to have raw power or get lucky. An important aspect of other bosses is that it is possible to prepare for them. Against enemies that use Fire attacks, you can bring Junpei and Koromaru (who resist Fire attacks), equip Fire-resistance items, and equip a Persona on the hero that resists Fire, which helps even out the battle against these foes. Against an enemy that uses Charm attacks, you can bring Personas that are immune to Charm and items to resist Charm. If an enemy has a Lightning weakness, you can bring Akihiko and equip a Persona with Lightning attacls. Unlike other games, doing this kind of preparation doesn't make the battle easy; it just makes the battle possible. This is because a lot of these abilities are restricted to the point where you can't just give them to every character, giving the team an incomplete defense that helps, but is still vulnerable. One of the strnegths of this game is that most boss battles are won or lost in preparation, and because of this the few bosses that you can't prepare for are simply overwhelming and unfun.
Anyways, the reason I delayed in making this post for so long is because I wanted to see the bosses in The Answer before making final conclusions, and I am glad I did so, because the boss battles of the Abyss of Time are even better than the Tartarus bosses. A major limitation of Tartarus bosses is that they come in homogenous groups. If you are attacked by a group of bosses all at once in Tartarus, you are guaranteed to fight a group of three identical enemies (all bosses other than the last one in each section are like this). This is more interesting than fighting single enemies all the time, but it is nowhere near as interesting as the non-homogenous boss groups in The Answer. In addition to making the "what is it weak to?" game more important, this adds prioritizing targets to the realm of strategies the player needs to consider (I never used the "Assign Target" command once in the main game, but it is important in The Answer). You have to choose between attacking weak enemies (who might get replaced by a bigger enemy's summon skill), killing a larger enemy, trying to kill them all evenly, killing the healer first, killing the status-effect monster first, etc. Often, even figuring out which enemies are the easiest to kill can be tricky. All of this makes good tactics even more important, and makes these battles very fun. What is more, The Answer doesn't seem to contain any unfair bosses that can't be fought with good preparation and strategy (with maybe one exception, but it doesn't use Megido), making the whole thing a lot more fun. I have not enjoyed battles like these in a console RPG for far too long.
Before I finish up, I want to make a special mention of The Reaper. That was one of the best optional super-boss battles I have ever seen. It was both incredibly difficult, but because tactics play such an important part of the game I managed to defeat it about 20 levels before I probably should have thanks to a good strategy and a little luck. That kind of thing is what I consider to be the best possible sign that a game is giving the right type of challenge.