Maximum height of a node in a red-black tree

If we have a node in a red-black tree with a black height of 3, what is the maximum height allowed for the node?

To maximise the height of that node, you'll want to create the longest downwards path possible with only 3 black nodes in it (excluding the starting node). As in red-black trees a red node cannot have a red child, the best you can do is to alternate the colors on such a path. Also, the path must end with a black node, since all (NIL) leaves are black. So the longest path with 3 black nodes is:

node (black, but not counted in "black height")
   \
   red 
     \ 
     black #1
       \
       red
         \
         black #2
           \
           red
             \
             black #3/NIL

The height (= length of path) is 6. It cannot be longer. So that is the maximum height of a node with a black height of 3. In general the maximum height of a node is the double of its black height.