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counter for stories, episodes of TV series, etc.
「話」 has another reading with different meanings.
今日は第三話まで見た。
I watched up to episode three today.
このドラマは全十二話だよ。
This drama is twelve episodes total.
最終話の展開が気になって仕方がない。
I can't help but wonder about the development in the last episode.