The story
From any + percent (%), denoting that any percentage of the game could be completed in order to get to the central goal.
Etymology from Wiktionary, CC BY-SA 4.0.
any%
Today it means: A category in speedrunning which pertains to the goal of a game's mere completion, i.e. getting to the main concluding goal as quickly as possible, regardless of how much is unlocked or what techniques are used. Full definition →