Gandalf / Saruman duel
One thing bugging me about the movies. When saruman is fighting with gandalf, he takes his staff from him and uses it with his own to send gandalf to the top of orthanc. Later when he gives gandalf one last chance to join him and he jumps on gwaihir he still does not have his staff, yet at rivendell he does. How the hell does he get it back?!
In the books, Gandalf is explicitly carrying his staff when Gwaihir comes for him (in Frodo's dream).
In the movies - good question. Maybe Gwaihir made a little detour to Saruman's staff collection, or maybe it isn't the same staff. I always got the impression that the staff changes slightly over time, "opening" its upper knot (compare for example the staff in Moria to the one he carries arriving in the Shire; but then again, it might also just be the perspective) - maybe it was actually replaced. Also, look at Gandalf's staff in the Hobbit - its knot is all closed, and the surface looks quite a bit smoother.
The elves also managed to find/produce a new staff for Gandalf the White, so maybe staffs aren't that unique at all.
Yes, he did have a staff when he arrived; in the Unfinished Tales, it says "and last came one who seemed the least, less tall than the others, and in looks more aged, grey-haired and grey-clad, and leaning on a staff."
The description, and also his names among Dwarves and Men (Tharkûn and Gandalf, both meaning "person/elf with a staff"😉, however, make it sound like he was the only one of the five Istari to bring a staff with him (and like he used it mostly as a walking stick).