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An Inn is a place where Mario and his Partners, for a small fee, can go to sleep so they heal their HP, FP, and, in the Paper Mario series, SP. The price varies by location; inns later in the game generally have higher prices. They appear in most of the Mario RPGs and games with similar elements.
Similarly, Paper Mario: The Thousand-Year Door and Super Paper Mario feature several inns which throw in an item as a bonus. In the former, if Mario sleeps in the Inns 50 times, a Cameraman Lakitu will appear, congratulate, and give him 200 coins.
Paper Mario's Toad Houses served the same function, with the main difference being that they were free of charge, and in Paper Mario: The Thousand-Year Door, the player could stay for free if Mario had an Inn Coupon in his possession.