According to this article on nytimes.com they have found a group of monkeys that are using syntax. Syntax is something we have long considered uniquely human. The closest thing I've ever seen to another animal using syntax is sign-language Koko using "bean ball" to describe a pea. But that's really compounding, not syntax.
I imagine there are other animals besides us and Campbell's Monkeys that use syntax in their communication, but it's yet to be seen. The article suggests whale calls are a possibility. I would love to see a study on that with more conclusive evidence. I always imagined whale songs to be the closest real equivalent to the mythical language of the birds.