8 the first time he comes out to pitch, 5 at the top of every subsequent inning - none, if he leaves the mound and legally reenters or pitches again.
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Well yes if they want to. If they refuse to they have a greater chance of getting hurt if they don't warm up.
As many as he needs. If a pitcher replaces another pitcher whom the manager just pulled, then the incoming pitcher gets a maximum of 8 pitches.