What is the ruling on taking an inexpensive watch found in the Grand Mosque (Masjid al-Haram) in Mecca, using it for the duration of one's stay, and then disposing of it? And is there any expiation (kaffarah) required, regardless of whether the finder was in a state of ihram or not?

A broken watch that has no value is considered among the minor items that may be picked up and utilized as long as its owner is unknown. It is better for someone to benefit from it rather than it being discarded. The state of ihram of the picker or the location of picking it up (Mecca) does not affect its ruling.