I can think of two ways a thought could be unthinkable:

1. It could be uncomputable, since it seems likely to me that we can't think uncomputable thoughts. (Not a rigorous scientific hypothesis, but it seems more likely than not to me.)

2. It could be uncomputable in a human lifetime. I think these already exist today: can anyone really be said to thoroughly understand a proof generated by an exhaustive computational search?

One can play around with the semantics of what we mean by "unthinkable" and "thought," but I think "computable" at least provides one interesting start to an exploration of this concept.