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A coworker of mine who used to work in London came to be pretty familiar with the steps of Covent Garden tube station during his commute ages ago. I've climbed it once or twice myself and it's kinda cool; I like 'x steps to go' signposts along the way.
So that's the most amount of stairs. It's certainly not the deepest tube station, let alone subway; there's Hampstead station where you're not even allowed to use the spirtal stairs unless there's an emergency, and of course stations around the world as other anons here have mentioned and more.