If we're going to take items and stat boosts into account, let's actually take them fully into account, under current rules and mechanics.
Max HP is of course 714, and nothing messes with that so there's no way to improve on the established result there.
For Attack, we can do something different. While it will involve a Shuckle with maxed out Defense and Power Trick, Shuckle isn't the one who will be getting the maximum Attack stat. Instead, after using Power Trick, Shuckle will use another move--Power Split--to split its attack with a Pikachu. Meanwhile, since this is a pretty time-consuming process, Pikachu can use some high-PP move like Agility purely to waste time.
If Shuckle started with 614 Defense, which was later moved to Attack, and Pikachu can only go as high as 229, then after the split they both have a raw attack stat of 421. At this point Shuckle will have to switch out and back in, which returns it to its normal status of 614 Defense and some low number in Attack, but now it can Power Trick and Power Split again. Each time through, Pikachu's attack stat will be raised, to 517, then 565, and finally after nine loops of that, Pikachu will have a raw attack stat of 613--one point below Shuckle's own starting maximum Defense. It's not possible to do any more productive Power Splits, because splitting 613 with 614 just rounds down to 613 again.
The reason for using Pikachu here, even at the cost of that inefficient Power Split technique that costs one point on the initial stat, is that Pikachu alone is capable of using the Light Ball, which uses the item slot to provide a 100% boost. (Cubone or Marowak would also work with the Thick Club, but all other species are limited to a 50% boost out of this slot, by using Choice Band.) So once we have the stat at 613 plus Light Ball, it's trivial to get to +6 Attack somehow, and then Skill Swapping to Huge Power or Pure Power is just a matter of having the right Pokemon on board to do that. But there's one final adjustment that affects the attack stat: after all this is done, have Pikachu's partner switch out to Cherrim, and use Sunny Day. This turns on the Flower Gift ability, which provides a 50% boost to the physical Attack and special Defense stats of both Cherrim and its teammate. All told, these boosts give Pikachu an Attack stat of 14,712.
Of course, part of "current rules" is that triple battles no longer exist; if you go back to an earlier rule version (such as ORAS) in which they did, you could easily throw in a second Cherrim, and all the other moves in the setup still work as described, boosting the Attack to 22,068.
Maximizing Defense requires a greater sense of urgency. The only item for giving 100% Defense is Metal Powder, and that only works on Ditto, and only if the Ditto has not transformed yet. Since Ditto is completely incapable of learning other moves to waste time, you have to have ways of both making sure Transform always fails, and doesn't run out of PP before Ditto KO's itself with Struggle.
It's not especially hard to make Transform fail; the two easiest ways of doing so are by having it attempt to transform into a Pokemon that's actually Zoroark in disguise, or having the other Pokemon put up a Substitute before it can transform. To facilitate the setup here, I'd go with the Substitute method...specifically, Substitute on a Smeargle, along with Skill Swap, Cotton Guard, and Guard Swap.
While Smeargle and Ditto occupy one side of the field, the other side can consist of four Shuckles that switch back and forth between each other. Predictably, these Shuckles would all come with max defense, but they won't need to use Power Trick in order to move that into an attacking number; instead they can use Guard Split with the Ditto and immediately switch next turn to one of the other Shuckles. As with Power Splitting Pikachu, it takes nine Guard Splits to raise Ditto to the maximum possible starting number of 613, and with two slots working in harmony, they can combine to get all nine of them down by the 9th turn of the battle. Smeargle can continue to use moves like Substitute to waste time, and should also use Cotton Guard at least twice.
Once the Shuckles are done with their Guard Split rotation, one of their slots can be replaced by a Furfrou or Alola Persian, which Smeargle immediately Skill Swaps to get hold of the Fur Coat ability, and another Skill Swap next turn gives that ability to Ditto. Smeargle's final move is Guard Swap; note that this is not the same as Guard Split. Guard Split operates on the numbers at the raw stat level, while Guard Swap applies to common boosts. After using Cotton Guard twice, Smeargle will be at +6 Defense, and then Guard Swap will give that boost over to Ditto despite it being a blob that doesn't have any boosting moves of its own. Between that, the 613, the Fur Coat, and the Metal Powder, Ditto now has a Defense stat of 9808, and even a few turns of Transform PP left with which to enjoy it!
Special Defense gives us the Deep Sea Scale as the only 100% item, and that works on Clamperl. Fortunately, Clamperl has moves that don't involve transforming, and that have more PP. Still, the same template of a bunch of Guard Split Shuckles works here, with the exception that this time they should be focused on Special Defense, not Defense.
9 Guard Splits is again the magic number for reaching 613. Clamperl can learn several moves, but none of them give itself boosts to that particular stat, so it'll need someone else to use a boosting move and Guard Swap it over. Unfortunately, there's no ability that can be Skill Swapped that has any effect on Special Defense, but there are two other things to consider. One is that if you go ahead and use Skill Swap anyway, to let Clamperl have Color Change as its ability, it's easy enough to turn it into a Rock type. The relevance of that is that while Sandstorm is up, all Rock types (regardless of how they got that type) get an inherent 50% boost to Special Defense. Alternatively, Flower Gift mentioned above also applies to Special Defense, but only while Sun is up. You clearly can't have multiple rivaling weather conditions active at the same time, so these two are mutually exclusive and whichever one you pick, it results in a Special Defense stat of 7356. Or going back to older rulesets, the Sandstorm bonus is unchanged in triples since it can't apply more than once, but the Flower Gift version of the bonus can be applied multiple times with two partners available, so you might as well pick that one, bumping it up to 11,034.
Special Attack is the relative laggard here. This mainly draws from the fact that Power Trick only exchanges the physical stats, not the special ones, and there is no special equivalent at all (but Power Split does exchange both attacking stats regardless). This means there's no way to get a 613 or 614 into this column as the raw stat; the best you can use is 535 from Mega Mewtwo Y. Item-wise, the Light Ball works again here, or else Clamperl can use a different item called Deep Sea Tooth.
Now, Mewtwo doesn't actually learn Power Split, but it does learn Mimic. This suggests that Mewtwo can be opposite of Pikachu, while the other half of the field can have two irrelevant Pokemon where one of them uses Power Split on the other. Then Mewtwo uses Mimic to learn Power Split, uses that move on Pikachu, then switches out and the cycle repeats (to include having to use Mimic again). It's complicated, but as usual 9 is the magic number of loops to get Pikachu's starting stat up to 534, one point less than what Mewtwo innately had after going mega, and both Power Split and Mimic have enough PP to allow that.
There's nothing like Huge Power for the special side, but at least there is Plus/Minus. Either Plusle or Minun can use Entrainment to pass its ability onto Pikachu with ease, and then when both Pokemon on a side have one of the Plus or Minus abilities, they both get a 50% Special Attack boost. It's better than nothing, and enough to lift Pikachu's Special Attack to 6408.
Finally, we have Speed. This is the fastest one to set up, and also the one with some caveats to it if you're just going for big numbers. Here the move worth building around is Speed Swap, which (contrary to the name) works nothing like Power Swap or Guard Swap, and doesn't work quite like Power Split either: instead of adding up the two underlying numbers and averaging them into two equal parts, it simply exchanges those underlying numbers.
In this case, I have a previous battle from the game itself, with the sole purpose of achieving this high speed, and it was recorded on video here. Basically, Accelgor uses Baton Pass to bring in a Ditto while it has +6 Speed, and that Ditto is immediately hit in the same turn with Skill Swap (changing its ability to Surge Surfer), Speed Swap (replacing its raw speed with 504 from Deoxys), and Switcheroo (giving it a Quick Powder), while Electric Terrain is in effect and both sides have Tailwind up. After all those boosts, Ditto has a very short-lived Speed of 16,128, at least on paper.
The reason I say "on paper" is that in this case, we know the game's accounting is different, and that video itself serves as proof. Instead, what happens is there's a hard speed limit (or two) imposed: any Pokemon that would have Speed greater than 10,000 gets limited to just that 10,000 figure, with anything extra being cut off. Furthermore, after that's done (along with a possible extra step to account for the presence of Trick Room), any Pokemon whose Speed is at least 8192 gets 8192 subtracted for the purpose of turn order. Together, this means that any speed greater than 10,000 will be treated as if it was only 1808 (for turn order purposes, but not for Gyro Ball and Electro Ball). You can see this mechanic in the video, where Swoobat (whose speed at that point was only 2512) managed to go before the "really fast" Ditto on the final turn.
Under a reckoning that does respect those known limitations, the ideal speed would be 8191, but it is impossible to land directly on that number with any combination of multipliers (unless Trick Room is in effect). Instead, the best you can do is 8190, which represents a speed of 455 at +6, paralyzed, with the Quick Feet ability, a Choice Scarf, and Tailwind in effect.
