We humans can take an entire proposition and give it a role in some larger proposition. Then we can take the larger proposition and embed it in a still-larger one, creating a hierarchical tree structure of propositions inside propositions. Not only did the baby eat the slug, but the father saw the baby eat the slug, and I wonder whether the father saw the baby eat the slug, and the father knows that I wonder whether he saw the baby eat the slug, and I can guess that the father knows that I wonder whether he saw the baby eat the slug, and so on. Just as ability to add 1 to a number bestows the ability to generate an infinite set of numbers, the ability to embed a proposition inside another proposition bestows the ability to think an infinite number of thoughts.I've seen many similar such examples, and they all say that because we can recurse, we can think (or generate) an infinite (I think they mean unbounded) number of thoughts (or sentences).
---Steven Pinker, How the Mind Works, pp124-5, Penguin, 1997
Funnily enough, the examples given always stop around the depth shown above. That's because although we can recurse 4 or 5 or 6 deep, we can't (well, I know I can't) go much deeper. My stack overfloweth.
Unboundedness requires us to be able to recurse to a depth much much more than 6. It requires any depth: to a hundred, to a trillion, to a googolplex, and beyond. Clearly, it is nonsensical to think that we can do this.
I believe that some of the writers who come up with such statements about unbounded recursion think we need it to get a sufficiently large number of sentences. But even without it, there is no problem that we might run out of thoughts. Combinatorics is quite sufficient to give a ridiculously huge number of possible sentences. "The small purple octopus frowned thoughtfully as it slowly drank cold green tea from the heavy pewter mug held gently in its fifth tentacle" -- I bet that's never been said before! And the number of sentences that follow just this one particular, rather simple, structure is mind-bogglingly huge (even the number that actually make some kind of sense).
So let's dispense with the idea of "an infinite number of sentences thorough unbounded recursion", please. We don't need to go that deep.