No, double or multiple contractions are not formal.
While some style guides support the moderate use of common contractions, even single contractions are sometimes discouraged in formal writing. See MLA style on contractions and this roundup of views on contractions.
Edit to address some of the points in the comments:
In formal writing, it is appropriate to use contractions if you are quoting a line of text or speech that contains contractions, or if the topic you are discussing is the use of contractions.
O'clock is standard and formal.
Diacritical marks in words like Qur'an, Hawai'i and Xi'an are not contractions and are not discouraged.
Is “cannot” a contraction of “can not?” Should you stick with “can not” in formal writing?
It is a contraction of "can not," but the contraction is perfectly acceptable in formal writing. Moreover, the long and short versions are no longer exactly equivalent in meaning.
I cannot do X.
"Doing X is not within my power."
I can not do X.
"Not doing X is within my power."
More on reddit.comHelp me understand how “aren’t” is a contraction in this type of usage: “You’re him, aren’t you?”
Was always told to not use contractions like "don't" "couldn't" "wouldn't" - should I/can I use these if it's a first person teenage PoV who would realistically use these?
Why do contractions sometimes work and sometimes don't? Like in this sentence: "I want to be a doctor like you're." Is there an official rule for this?
Videos
No, double or multiple contractions are not formal.
While some style guides support the moderate use of common contractions, even single contractions are sometimes discouraged in formal writing. See MLA style on contractions and this roundup of views on contractions.
Edit to address some of the points in the comments:
In formal writing, it is appropriate to use contractions if you are quoting a line of text or speech that contains contractions, or if the topic you are discussing is the use of contractions.
O'clock is standard and formal.
Diacritical marks in words like Qur'an, Hawai'i and Xi'an are not contractions and are not discouraged.
I agree with Katy's answer that in quite formal contexts even individual contractions are frowned upon. But, since you ask about double contractions, you may be interested in writing that is informal enough that contractions are acceptable, but still somewhat formal. And, generally, your intuition is valid that doubling up contractions increases the informality. For instance, a news article about trade negotiations is generally written in a fairly formal style, but contractions are often shown in quoted speech. But, if someone said "wouldn't've" out loud, that would show up as "wouldn't have" in the article.
On the other hand, the way you write dialogue in a novel is part of your style. You may want to convey more about how the speech is being delivered than what comes across in standard, cleaned-up prose. I would still advocate restraint. Complicated contractions, like dialect respellings, tend to stand out and can distract the reader.