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I am just curious about exactly how high the total base power of Beat Up could potentially go.

For this question I want 6 different Pokemon, so the Species Clause applies, and a max of 1 Mega, but everything else goes.

If you don't know what Beat Up does, then check the link, or for some more information look at Bulbapedia

I assume Choice Banded Weavile is the best user, but if there is a better one, then let me know that as well.

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You would need 5 Pokemon that have the highest Attack stat, and the Pokemon using Beat Up.

Banded Weavile would be the best Pokemon in this case. Let's do all the calculations:

Weavile: 120 / 10 + 5 = 17
Mega Mewtwo X: 190 / 10 + 5 = 24
Deoxys-A: 180 / 10 + 5 = 23
Kyurem-B: 170 / 10 + 5 = 22
Rampardos: 165 / 10 + 5 = 21.5 -> 21
Slaking: 160 / 10 + 5 = 21

17 + 24 + 23 + 22 + 21 + 21 = 128 BP

With STAB: 128 * 1.5 = 192

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thanks. Choice Band stacks on top of that doesn't it?
It sure does.
Would Mega Mewtwo's stats count if he's not mega evolved? He'd have to be the one using beat up so you could mega evolve him right?

Might I suggest:
Mega Tyranitar: 120/10 + 5 = 21.4
Kartana: 181/10 + 5 = 23.1
Kartana: 181/10 + 5 = 23.1
Kartana: 181/10 + 5 = 23.1
Kartana: 181/10 + 5 = 23.1
Deoxys: 180/10 + 5 = 23

21.4 + (23.1 * 4) + 23 = 136.8
with STAB: 136.8 * 1.5 = 205.2

Mega Tyranitar has dark typing.
You get four Kartana's in the UB quest in Sun (heck you could make this even higher if you can get someone to trade you a fifth one (or all five in the case of Moon players)).

If you can't get four Kartanas (because you play Moon and don't know anyone that generous) you could also run:
Mega Tyranitar: 120/10 + 5 = 21.4
Kartana: 181/10 + 5 = 23.1
Deoxys: 180/10 + 5 = 23
Rampardos: 165/10 * 5 = 21.5
Rampardos: 165/10 * 5 = 21.5
Rampardos: 165/10 * 5 = 21.5

21.4 +23.1 + 23 + (21.5 * 3) = 132
STAB: 132 * 1.5 = 198

You can breed Rampardos so why not run more than one instead of using Slaking.