tolkiens legendarium - How much of Gandalf's power is tied to his staff?

Before Gandalf freed Théoden from Saruman/Gríma's hold, he refused to give away his staff.

Háma: I cannot allow you before Théoden King so armed, Gandalf Greyhame... by order of Gríma Wormtongue.
[Gandalf nods, and they hand over their weapons]
Háma: Your staff.
Gandalf: Oh... you would not part an old man from his walking stick?

After that, while he was breaking theirs hold on him, Gríma shouted to the guards

His staff! I told you to take the wizard's staff!

But what if he was denied the use of his staff? How much of his power is tied to his staff? Is he harmless/defenseless without his staff? If his staff is taken by an other magically capable being, how much of Gandalf's power would it have?

Answer

I couldn't find anything directly related to this in canon, but circumstancially, the staff seems to be needed to help express the magic of the Istari, but is not required/critical to it.

In this way, it seems somewhat akin to wands in Harry Potter - a good wizard can do magic without a wand but the wand helps focus the magic.

On the side of the staff helping magical usage:

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  Gandalf uses his staff to break the bridge on which the Balrog is advancing

  
  


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  Gandalf's staff is used to send a signal to the Giant Eagle to help escape from Orthanc.

  
  


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  Gandalf the White breaks Saruman's staff after asking to surrender it.

  
  

  This is a really weak and inconclusive data point, since it may have had more of a symbolic/moral meaning than practical.

  
  

On the side of the staff not being critical:

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  As per FotR, Gandalf was allowed by Saruman to keep his staff when captured (see the quote below pertaining to Frodo's seeing Gandalf's escape in a dream). If the staff was so critical to his ability to do magic, surely Saruman would have taken it from Gandalf!

  
  

  
  

  In the dead night, Frodo lay in a dream without light. ... The figure [Gandalf] lifted his arms and a light flashed from the staff that he wielded. A mighty eagle swept down and bore him away...