Intelligence and consciousness are emergent in nature. In this regard, I think it’s extraordinarily difficult to understand one level from a different one, particularly moving upward. In this regard, let’s expand on the example in the OP of talking to a dog. From our perspective, we do a lot to give human traits to dogs in our minds; that is, we assume certain cognitive and emotional responses to stimuli that just aren’t there. And from the dog’s perspective, it’s not that he just can’t, in the way an unintelligent and/or uninformed person might not be able to, but he can’t even grasp the concept of having such a conversation. I might guess that a dog has some concept of the idea that we have aspects that are beyond him, but all he can do is relate it to his own experiences, and so humans just fall into the role of alpha as the closest they can get.
In this regard, we know there are some things that are beyond us, like perhaps actually grasping a fourth spatial dimension, seeing outside the visible light spectrum, hearing outside the typical range, an ability to sense electromagnetic fields, etc. But that’s not intelligence, that’s just a different set of senses that connect us to the world. I think we’d see that an intelligence that is at least one whole level beyond us would be, at best, on the very edge of our ability to understand what it is they’re able to do that we’re not, and quite possibly just straight up beyond us.
But as someone mentioned upthread, I think there’s at least one more level, and it is the idea of the connected human mind. Compare to hive animals like bees and ants. An individual member of the colony isn’t able to do anything remarkable compared to other insects of similar complexity, but when you put the colony together, it’s able to perform remarkably complex behavior, and this behavior is fully beyond the understanding of any one member, it only knows it’s particular role. In fact, there’s some behaviors that colony animals are able to perform that we can’t even understand; for instance, I recall reading that research had shown that bees are able to solve TSP in a more efficient way than we’ve been able to figure out with modern computers.
So, what if the next level is what we’re approaching now with the use of the internet, social media, and similar technologies in the future. What if we’re able to combine the minds of hundreds, thousands, or millions of minds in a way to solve problems as far beyond that of a single human mind as the sorts of problems a hive solves that are beyond a single member. We already have the concept of the technological singularity, what if there’s something similar to that with consciousness and intelligence, perhaps tied to technology, and it only appears as a singularity because it’s beyond our current ability to understand.
So, sure, I’m reasonably certain there’s at least one more level of intelligence and consciousness beyond us. I’m also reasonably certain that if there’s any sort of interstellar or even intergallactic race of aliens out there, they’re probably at that level or beyond. Maybe that’s also part of why it seems so far beyond us, precisely because that sort of problem is beyond the ability of a single human mind or even a group of human minds to solve, but might be solvable when we’re able to think and operate at a higher level.