The potential number of rooms is probably either infinite or very, very large, and the actual number probably changes regularly. We know that the TARDIS can move rooms into storage, re-create old rooms out of storage (even if they haven't yet been created and stored), jettison rooms, surprise the Doctor with new rooms he didn't know about or old rooms he'd forgotten about…

If you want to know how many of those potential rooms are 'active' at any given time, probably not even the Doctor knows the exact count, but it's often been hundreds, and possibly a lot more than that. There have been a few times when this wasn't true, for different special reasons (when first built, when contaminated or damaged, etc.), but generally, the best answer is probably just 'lots'.Unknown, but definitely quite a lot. All the companions are assumed to have bedrooms, some have even been seen. There is a huge wardrobe seen in "The Christmas Invasion" as well as many other rooms mentioned an seen such as a library, a swimming pool room, a holding room, a cloister bell room, a kitchen, a sick bay, a scullery room, at least 7 squash courts, an art gallery, a 'Zero Room' and many more rooms.

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