In the book The Two Towers, Gandalf is puzzled as to how Saruman had been communicating with Sauron. Grima Wormtongue, rather conveniently, then throws Saruman's Palantir out the window of Orthanc, attempting to hit either Gandalf, Saruman, or both. Pippin picks it up, and Gandalf quickly snatches it from him and wraps it in a cloth. Clearly, he knows it is dangerous, but we later discover that he doesn't realize what it is.
Hours later, Pippin steals the Palantir and inadvertently uses it. Only then does Gandalf realize what it is. This would be understandable if Gandalf didn't know that Palantiri existed, or that there had once been a Palantir in Orthanc, or that the others had been lost long ago, but he reveals that he does indeed know all these things. His failure to immediately identify the Palantir is all the more confusing considering the fact that, as the group rode away from Orthanc, they heard Saruman shriek in horror, and Gandalf instantly knew that Saruman had just realized that Grima had chucked the Palantir out the window.
Gandalf's excuse for not figuring out what the weird crystal ball thrown from a wizard's tower was is weak at best - basically, he says "I was thinking about Saruman when the Palantir appeared, then I got tired":
But my mind was bent on Saruman, and I did not at once guess the nature of the Stone. Then I was weary, and as I lay pondering it, sleep overcame me. Now I know!
(The Two Towers, Book III, Chapter 11, "The Palantír")
He also says he is glad Pippin stole it, because if that hadn't happened, he would probably have used it himself (in his words, "I had considered whether or not to probe this stone myself to find its uses"), and if he had done so, he would have given Sauron far more information than Pippin did. This surprised me almost as much as Gandalf not recognizing the Palantir.
But I am still more flabbergasted by the fact that Gandalf didn't recognize the Palantir for what it was the moment he saw it, especially since the Palantir appeared while Gandalf was trying to figure out how Saruman had been communicating with Sauron. It seems to me that when he was wondering how a wizard was chatting with a mystical evil being and saw a crystal ball fly out of the wizard's window, he should have been able to put 2 and 2 together and make 4.
Is there an explanation for Gandalf's lapse more believable than the one he offers - "I was weary"?
Edit: For clarity's sake, I will lay out my case in as concise a manner as possible:
Gandalf knew that people once used glass balls to communicate;
and he was puzzled by how a guy had been communicating;
and Gandalf knew that the guy lived in a tower which once held such a ball;
and Gandalf knew that it was at least possible that the ball was still there;
and then a glass ball was thrown from that tower by the guy inside's lackey;
and Gandalf knew at once that the ball was potentially dangerous;
and then the guy inside saw that the ball was gone and shrieked;
and Gandalf instantly knew that the shriek was a reflection of the guy's displeasure at learning that the ball was gone;
and Gandalf is unimaginably wise.
In light of all this, Gandalf should, in my view, have been able to put the pieces together without a hobbit getting involved. I could have put the pieces together, and I am infinitely less wise than Gandalf. But the immeasurably wise Gandalf couldn't solve this mystery despite having the solution literally sitting in his lap.
See what I mean? The nature of the glass ball couldn't have been more obvious if it was written on the wall of Orthanc in letters 10 feet high.
I could understand Gandalf not immediately realizing Saruman had a Palantir if it hadn't been thrown at him and ended up in his lap, or if he hadn't been wondering how Saruman was able to communicate with Sauron when the Palantir was thrown at him, or if he was wondering about how Saruman was communicating with Sauron and the Palantir was thrown at him, but he had never heard of the Palantiri before. But he DID know all about the Palantiri, and he WAS wondering how Saruman was communicating with Sauron, and it WAS thrown at him, ad it DID end up in his lap. The pieces were all before him, yet he couldn't put this relatively simple puzzle together on his own until Pippin intervened and discovered exactly what the glass ball was.