This theory is going to include a lot of different points. But it involves three animals specifically. Those three animals all show different levels of intelligence that indicates they are more than base instinctual being despite ve some to a great amount of thought process. No, I will not include episodes involving scary stories as references. Because those are more of a way to show an alternate way things could have happened, and instead I will focus specifically on the main moments where animals are shown to have levels of intelligence not associated normally with animals.
First off, my favorite character in Gravity Falls, Waddles the Pig. He has multiple moments where he shows above-average intelligence, yet they are subtle in nature. The first is his interactions in the first episode he appears in. Think about his reaction to Mabel vs his reaction to Pacifica. His reaction to Mable is equal to a baby looking at a loved one. Yes, that would normally happen with a pig that was chosen for a prize. Yet, his reaction to Pacifica was odd. Like he knew the multiple timelines that had happened despite the reset Dipper applied each time. I find it odd that Waddles doesn't seem to be affected by multiple moments of what this show calls "weirdness". Since his first appearance is not the only moment this happened. He also shows an innate ability to comprehend human speech. Even though he can't speak it within the terms described above, he has been able to show that he understands and answers in other ways, especially in terms of enjoyment or for an odd reason boredom. The first is that this pig is willingly doing what Mabel asks him, with zero complaints or training. Remember he started in what could be considered farming circumstances, he was not meant to be a pet, but a prize. And as for most if Gravity Falls residents, that meant food. Also, it is confirmed he gets bored without Mabel. It seems normal, but he is known for leaving the shack when this happens. Mabel expresses concern to the point of leaving him with Stan. Not a person she knows to be remotely helpful, but all she had at the time. During the same episode, there is another moment that is lost on a lot of people. Immediately upon leaving the shack, this pig was assaulted by a pterodactyl. Yet, the string that was attached to him has an insane moment. It goes from the shack to the top of a tree, then heads into the woods. This points to Waddles climbing the tree, then being attacked by the dinosaur, and carried off. I have no explanation for a pig doing this feat, other than if he had some knowledge from previous attempts and possible successful ones due to the height. Knowledge a regular pig would not be able to keep or use if from a background in agriculture. The final point for Waddles having human-level intellect is the actions taken during his time in the Bubble. He seems to be able to use it for his own purpose to some extent. Since Mable made him her unofficial mode of transportation, you would think her actions would merit their escape using Waddles. Yet, Waddles seemingly saw far enough ahead to ready himself for an escape. Think about it, Waddles was not only at the point needed for an escape but also ready and equipped for the ride for an unknown amount of time beforehand despite the trial going on for some time.
The second animal that could have levels of intelligence beyond basic animal levels is the goat. This one has a bit less information than Waddles, but still gives a surprising view of the goat's actions throughout the series. First of all, I will head to the moment that Ford returns and tricks the government agents. Where did Ford place the USB? He gave it to the goat, almost a cryptic sign from the creator that this goat has been around for a lot longer than we know. I believe Ford had the goat during his Research Period and this goat is one of the ways he got rid of the lost pages restored after Weirdmageddon. So, this goat could actually know more than most characters in the show. Another moment actually helps us see the goat's emotions and thoughts in regard to the Pines Family in general. This goat has a grudge against the Pines Family. And it seems to go back to when Stan took custody of the shack. Since Stan is the only member of the family the goat is always butting heads with. Stan even tries to get rid of it just before Weirdmageddon, only to see it grow to enormous proportions and attempt to harm him in return. This is something animals would not be able to convey easily, unless it was a domesticated pet like a cat or dog. This goat showed at that moment that it would not do as Stan wanted, and can be seen throughout Weirdmageddon destroying the forest. Why the forest, however? I believe that the forest was a problem to the Pines Family, and this goat took it upon itself to protect the Mystery Shack by attacking magical creatures that would come to the shack to harm the Pines Family. This goat, however, is not the last animal in Gravity Falls with high intelligence.
This one is a stretch, but does anyone realize the raccoons are exhibiting an oddly teenage human mindset. This one has to be pulled from multiple references left throughout the show. Specifically, Old Man McGucket and Lazy Susan, who deal with them surprisingly often. It is confirmed by McGucket that the raccoons treat him with the same treatment as Robbie and his friends, and yes, it is confirmed the actions were done by raccoons. They are easily a conniving and mischievous group of animals. But McGucket also keeps one with him. Could they also have been an experiment done by McGucket for replacing memories? We know you can view memories from the gun to gain them again, but haven't seen the full extent of what can be done involving it other than reversing side effects on those affected through using your own memories. Yet, knowing McGucket's memories, he slowly lost both memory and function granted from each memory. Meaning, at some point the gun would be though of as a weapon and/or the raccoons would see one of the many memories and begin to gain said memories.
In conclusion, I sau, to me, Waddles is indeed of human-level intelligence is at least sub-human level intelligence, and those raccoons currently evade logic explanation for now.