Start with a Shuckle, who maximizes...not Attack, but Defense. The reason for this is because of a move called Power Trick, which will exchange those stats and move the high stat into the Attack column. After using that, Shuckle will use another move--Power Split--to split its attack with its doubles partner: 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.
If Pikachu had some way of being able to use Transform (such as in hackmons), you don't need to use the iterated Power Split process: you can simply have Shuckle use Power Trick once, then let Pikachu transform into it and get the full 614, whereupon it still gets to benefit from Light Ball even though it transformed into a Shuckle, because that item, unlike the Ditto-specific items of Quick Powder and Metal Powder, checks the pre-transformed state to determine whether it should apply, rather than the post-transformed state. Throw everything else on top of this and if there were a legal way to get Pikachu/614 at the same time, all the bonuses above that would raise the final stat to 14728 in doubles, or 22092 back when triples existed.
