What languages are available that promote both object-oriented and functional programming? I know that any language that supports first-class functions can be considered fun
You're really asking the wrong question. Your question is premised on there being a distinction between "OO" and "functional" programming. This distinction isn't interesting or relevant. In fact, according to the "supports first-class function" criteria, even Java is functional.
But you hit the nail on the head with "purely functional". That's the interesting bit. What languages offer you referential transparency and purity like that? Well, most of them, if you're very careful. But not many of them actually guarantee that your functions are pure. I can only think of a couple of languages that offer you that.
One of them is Haskell. In Haskell, you write programs that are purely functional from beginning to end. Side-effects are delegated to a data structure called IO, and state is handled by passing it through pure functions or encapsulating it in monads. So you have your "debugging heaven" where only a small portion of your code interacts with global state and the rest of your program is pure, and purity is enforced by the language.