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the highest attainable ability power by a playable champion is 1874, on Mordekaiser. To reach this amount, he must:
~have a fully stacked Mejai's Soulstealer and five Rabadon's Deathcaps.
~have a full page of Force runes.
~ have the Summoner's Wrath mastery and maximum ranks in the Blast, Mental Force, and Archmage masteries.
~possess the Exalted with Baron Nashor buff.
~have consumed an Elixir of Brilliance.
~have Ignite on cooldown.
~be affected by a Will of the Ancients aura.
~be affected by the auras of nearby Taric's Radiance and Sona's Hymn of Valor abilities, and under the effects of Lulu's Whimsy.
~have killed an enemy Karma with the same setup with Children of the Grave