It’s can, can’t or cannot.
Can not does not exist.
-Or so says the corporate writing style guide I’ve been looking at. What’s more, I agree.
But why is it like this? What happened, etymologically and historically, to make this happen?
Is cannot actually something other than what it seems (the inelastic collision of the words ‘can’ and ‘not’)?
According to the OED, you can use can not.
You can follow the guide or you can not follow the guide; the choice is yours.
Under quite specific conditions, yes.
You can use “can not” these days, but it usually means something a bit different to “cannot”
Eg.
“Can you not smoke in here please?”
“Sure I can not smoke if it bothers you” (quite different implications to “I cannot smoke if it bothers you”)
One of our Dr Seuss books (One Fish Two Fish I think) doesn’t follow this style guide, which makes me wonder if the rule was different in the 50’s
Oh dear, Oh dear, I can not hear/Will you please come over here?/Will you please look in my ear?/There must be something there I fear
My husband reckons this is Just Plain Wrong but I imagine an extraordinarily popular kids book would make at least some effort to get their grammar right. So maybe it’s a fairly recent change? It is nice in terms of linguistic precision to have the two different shades of meaning available.
Valid point. We have do and don’t, but not donot. Did, didn’t, but not didnot.
Dunno - seems more likely that Seuss was doing his own thing - he’s kind of well known for it.
I think that cannot versus can not does express a distinction that other [verb]+not combinations don’t have.
Eg
“I can not go into the room” (I am able to refrain from going into the room - but I might not choose to)
“I cannot go into the room” (I am completely unable to go into the room - I have no choice)
Supposing we had a word “didnot” - what useful distinction could we make between that and “did not”?
“I didnot go into the room” (I failed to perform the action of entering)
“I did not go into the room” (I performed the action of refraining from entering)
They mean the same. There’s no useful distinction to be made, so we don’t need the two different terms.
Ok, I buy that. So to the OP, the answer is that the word can is what is special.
That’s an interesting point - and I can’t think of an example to break it, but is that the true reason for it?
There’s also the fact that “cannot” rolls off the tongue euphoniously much better than “didnot” or “willnot” (clunky IMO) or “donot” (spelt badly for its intended pronunciation). That may be why we have both “cannot” and “can’t” which, after all, do seem to have exactly identical meanings.
There’s also the fact that “can” ends with the letter “n,” which practically begs to be joined with the “n” in “not.”
ETA: I did not preview, so I missed Aspidistra’s highly intelligent and insightful post. 
<sigh> English is so frustrating to computer programmers.
Too bad we can’t say it in C:
I !can go into the room;
I can !go into the room;
There’s your problem. Corporate writing guides have nothing to do with English.
My speculation is that “cannot” is spelled as a single word because, as has been pointed out, “can” ends with the same consonant that “not” starts with, and this makes them more likely to blend together and be perceived as a single unit.
Keep in mind that this is purely a spelling convention, though. It reflects nothing in the grammar of English as acquired by native speakers.
As has also been pointed out, an advantage of this quirky convention is that it makes it easier to signal in prose when “can” is intended to take semantic scope over negation by spelling it out as two words: “can not”.
Other modals are similarly ambiguous, e.g. “may”, and for them, spelling is no help:
1a. You may not do it = It is not the case that you have permission to do it.
1b. You may not do it = You have permission not to do it.