Is it grammatically incorrect to use the word shouldn’t’ve? As in “should not have”. I’ve never seen this in print, but I use it all the time.
How much freedom do I have to make my own contractions, in general?
You wouldn’t be the first to use it. Google shows close to 2 million hits.
There’s nothing ungrammatical about it, but it generally regarded as informal and colloquially, so I would avoid it in formal pieces.
I’ve spoken it, but never actually written it out. It would seem like lazy writing if I were to see it in a letter or email.
The answer, as usual, is related to context.
Who is going to read it? Who, if anyone, is going to judge you/grade you/pass you/promote you because of it?
A double apostrophe is not accepted in formal English. It is greatly discouraged in “good” English: books, newspapers, magazines. Once you get into the realm of colloquial English, then you get to decide for yourself whether to use it. The majority of prose that most people write most of the time is colloquial. Message board posts are. But you probably would never see it in a column by Cecil or a staff column.
Of course, message board posts often contain prose that borders on unreadable and psychotic and illiterate. People will make judgments about you based on the way you write because that’s all the evidence they have to go by.
There are no rules about usage. Just contexts and what peoples’ expectations are.
Not even Fish ‘n’ Chips? 
(Mostly joking, but seen in newspapers at least).
You’ll find no shortage of “good” English newspapers and magazines using names such as “O’Reilly” in the possessive.
Do I really have to say a double apostrophe is the use of a contraction of two separate words? As in, oh I don’t know, shouldn’t’ve?
Fish ‘n’ Chips isn’t a double apostrophe. O’Reilly’s isn’t a double apostrophe. Fo’c’s’le isn’t a double apostrophe.
When I say that context is everything, the context of responding to the specific question posed by the OP is part of that context.
You’re right. That’s a triple apostrophe.
(FWIW, I would consider “Fish ‘n’ Chips” an example of a double apostrophe, and it still makes your point.)
I’m all for people who are creative and want to invent new useful forms and constructions, provided they do it intelligently.
In other words, “shouldn’t’ve” is okay. But stay far away from “shouldn’t of”. (Over 6 million hits on Google! Aaargghhh!)
Now you’re just trying to wind up Exapno Mapcase. 
OK, so the (formal language) preference appears to be avoiding multiple enclitics.
e.g. They’ll’ve
Or, as in the case of the OP, an enclitic and an affix (that some people claim is an enclitic).
i.e. Should’nt’ve
In further news there’s a part of speech called a clitic. 
Merely finding that out (by initially looking up the Saxon genitive), makes being snarked at by Exapno Mapcase well worthwhile. (No, seriously… I learnt something new). 
Having no idea how convoluted and dazzling that article would be, and despite my tired and sleepy state, I tried to read it.
I shouldn’t’ve.
I’m an affix guy myself.
That’s a pretty dizzying article. I wish it had more examples in English, and didn’t rely on other languages so much.
Would “a whole 'nother” be an example of what they’re talking about?
I just wanna say that it all depends on the level of your writing. In really high style, even “wouldn’t” would be frowned on (I wouldn’t’ve ever used it in one of my journal publications, for example). But on the Boards, I see no problem (although the spell-checker does).
Commander Data would never use it.
You’d think so, but the actor (or possibly the writers) would quite often slip up. There are nearly a dozen instances of Data using contractions without any logical, story-internal explanation.
Oddly enough, the question “Are you an affix or clitic guy?” doesn’t come up as often as some similarly phrased questions.
Ah, but wouldn’t the world be a better place if it did?
Indeed, the article should’ve been written in English.