In the very first episode, we learn Max is naturally immune to hypnosis and brainwashing. The following episode, after Sam gets hypnotized himself (see Hypnotic Head), he gets himself some kind of hypnosis-proof hat. The former would explain perfectly well why in "They Stole Max's Brain!", after Sammun-Mak rewrites history and makes himself ruler of all creation, Max is able to recall what actually happened and recognize that history was rewritten. Why not Sam? The first thing Sam does upon discovering Max's brain was stolen and going on his noirish Roaring Rampage of Revenge is ditch his trenchcoat and hat. Nice job screwing yourself there, Sam.
The same trick is pulled in "Beyond the Alley of the Dolls." One of the Samulacra/Dogglegangers steals Sam's hat early on, leaving him without it for the majority of the episode and leaving him vulnerable to Charlie Ho-Tep's mind control.
Related to the above, the process for undoing hypnosis and brainwashing involves "destroying the inturder (the brainwasher) in their mind". Given how incredibly violently psychotic Max is at the best of times, it's unlikely ANY being in his mind doesn't get killed constantly, not to mention a foreign one, so of course he's immune.
During the text adventure in "Reality 2.0," you go down to discover the Shambling Corporate Presence. Since they went down, they would presumably be entering the underground sewers. It's not until the next season that you learn that the Shambling Corporate Presence is a demon, and the gateway to hell is directly under their building, in... wait for it... the sewers. So it's not actually that improbable that an entity like the SCP would be there. Not that they'd know that; since the graphics weren't working, they wouldn't be able to tell where they were at the time, which is why they didn't notice the gateway-to-hell thing earlier.
The Devil doesn't factor in to "The Devil's Playhouse" at all, except as an extra in the finale. The title is later explained to have come from the saying "Idle hands are the devil's playthings." If this is true, then an idle mind would be the devil's playhouse. The name of the toychest itself, The Devil's Toybox, most likely refers to the fact that the Toys of Power are being used for meaningless purposes by the aforementioned idle minds. As Max's neglected Superego, The Narrator explains that this is why he referred to it as his playhouse in the beginning - he's an entity of Max's mind, but Max only uses his incredible powers for the most pointless things.
In "The Tomb of Sammun-Mak," the player has to try and get Charlie Ho-Tep from Nefertiti; but she's unwilling to give him up, because she claims tobe in love with him. Then, two episodes later, we learn that Charlie's actually sentient, making the concept of a romance between them a hell of a lot less creepy and pathetic, though still kind of improbable.
In the finale of 305, we learn that the Max that appears in the time elevator is alone because Sam turned into a giant monster after developing electromagnetic powers in an alternate timeline and he left to presumably find Sam at some other point in time. How did this occur? Remember the Time-Traveling Mariachis from 204? On-board their ship, you meet a past version of Sam & Max from 102, where you get their recording contract, which they need to get on Myra's show, instead giving them the screwdriver. Thus, you presumably forced them to use THAT instead of what they originally needed, causing a time paradox that altered their timeline, thus causing the events that lead to Max being the one to survive, rather than Sam.
Unintentional, but Max's super-ego is tired of being unused and neglected. He is voiced by Andrew Chaikin, who voiced Max back in 101. Seems they both went "unused" for a while.
Given everything we learn during the events of Season Three, it makes an oddly large amount of sense that back in 103, a moleman became the leader of the Toy Mafia.
"Please poke the eyes to see an ancient and brutal civilization spring to life."Wait, you mean the one that occurs after Sammun-Mak gets close to the Toybox after Papierwaite breaks it off with Skun'ka'pe? You know, because his minion assaulted Sal after Sal poked him in the eye?
This may not be intentional, but at the very ending of 305, there are two possible explanations for Max's head catching on fire, even though the Narrator stayed behind to fix it. Either his efforts failed, or what he said to Sam was just a lie, his grudge against Max was intact and he sabotaged the situation. The latter seems a bit more likely, considering two lines. When Sam asks what the Narrator is doing in Max's brain, the Narrator replies that he is Max's brain. And right before Max's head catches fire, one of the spores complains that "my head is killing me".
How did Papierwaite stay alive for so long? Generic dark magic, right? Not quite. He says to Max in 303 that the secret of his apparent immortality is too bizarre for the mortal mind to comprehend. Sure, dark magic can probably cause insanity. But in 305, he and Yog-Soggoth are exposed to radiation, and Yog-Soggoth comments that they'll have to wait out the half-life, making it necessary for them to stick together another hundred years. As long as Yog-Soggoth and Papierwaite are stuck together, he cannot die. And elder beasts are infamous for making mortal minds go mad...
If you try to open Papierwaite's door from the inside of his office in 303, you get this exchange.
Sam: It's locked. What sort of paranoid weirdo keeps his door locked from the inside?
Sammun-Mak: I locked all my slaves in my birthday pyramid after they finished building it.
Sam: Papierwaite's not a slave!
Sammun-Mak: Isn't he?
Given that Sammun-Mak refers to Papierwaite as "a lowly scribe" if you examine his books, and Sammun-Mak has a general tendency for viewing everyone as beneath him, this statement seems like just another joke. But it also serves as foreshadowing that there's another presence waiting in the wings.
After meeting with Specs making graffiti of Brady Culture, Max says he needs to paint him next. Guess what Specs is seen doing in the ending?
Likely unintentional example - In Sam & Max: The Devil's Playhouse, Max sees strange images when looking through First Person Camera that initially start off psychedelic and cheerful in Episode 1 (floating wrenches, Sam's tie being replaced with a fish, a beach ball bouncing along the stools at Stinky's Diner), but become less common and more 'solid', thus more difficult to distinguish from reality, as the story progresses. Word of God is that these images are echoes of Maxes from other universes going on adventures, since they share a sort of Hive Mind. The more you think about it, the worse it gets:
Why does the frequency of hallucinations decrease? The logical conclusion is that more and more alternate universe Maxes are dying. Considering the plot of The Devil's Playhouse is partially about a component of Max's psyche trying to destroy Max in order to protect people from his (own) psychosis, this suggests there are increasing numbers of worlds where he succeeded.
Certain specific hallucinated images are actually pretty disturbing when you think about their implication. In Episode 4, Max 'sees' Sammun-Mak in Papierwaite's office, surrounded by hovering Lugers pointing at his head. This implies there is another world where Max failed to conquer Sammun-Mak, and the false reality he created still existed - and in that reality, the world was under an oppressive rule where the Moles were ghettoised, and Max was a disembodied brain.
The Narrator is the manifestation of Max's superego, meaning that he's simply the personification of a part of his subconscious. After the reveal that he was season three's Big Bad, Sam demands to know what his goals are, and he explains that all he wants to do is to "let it all end." This implies that Max has suicidal tendencies. When you realize this, it makes you wonder what, exactly, happened to Max to make him like this. He's obviously unstable, it's been implied several times that his childhood was a traumatic one, he suffers from night terrors, he's been to therapy in the past...maybe he's more messed up then we thought at first.
At the finale of season 1, the entire population of Earth is hypnotized into thinking they are and acting like Max. Think of the implications of that for a second, there's BILLIONS of people acting like that completely psychotic little bunny. The fact that anybody is left alive for another season to happen is surprising.
Max was president during Season 3, meaning the country is left without a president after he dies. And even if the Past Max takes over the role, the president still became a gigantic monster that nearly destroyed New York, meaning he's probably in for a difficult reelection campaign.