When Frodo, Pippin, and Sam were traveling to Rivendell, they slept under a tree the first night. Tolkien writes about a fox that wonders to himself why three hobbits were sleeping under a tree. Why can the fox think for himself? This isn't Narnia where animals can talk, but this is the only time I can think of where an animal thinks for itself in Lord of the Rings. Here is an excerpt from the book:
A fox passing through the wood on business of his own stopped several minutes and sniffed. "Hobbits!" he thought. "Well, what next? I have heard of strange doings in this land, but I have seldom heard of a hobbit sleeping out of doors under a tree. Three of them! There's something might queer behind this." He was quite right, but he never found out more about this. -Fellowship of the Ring
What other strange thing could the fox have heard of and who or what did he hear them from?
Could the Great Eagles be considered a true animal according to this? They have no equivalent in the real world of the same size, so I don't count them.