Won’t actually has a pretty interesting and complex history. Ultimately it does come from a contraction of will and not, but it all happened in a rather roundabout way.

It all started off with the Old English verb willan/wyllan, meaning to will, wish, or want. Even in Old English it was used occasionally to denote a future intent. “Ic wille gan” could mean “I want to go” or “I will go”, depending on context.

Now, the thing about negatives in Old English is that they were often reduced:

na(w)ðer = nahwæðer = ne + hwæðer
neither = not + whether

næfre = ne + æfre
never = not + ever

nabbað = ne + habbað
haven’t = have + not

We nabbað naðor ne hlaf ne wæter.
We have neither bread nor water.

Not comes from naht via noht. Related to nawiht meaning naught, it originally meant in no way, but came to be used as an emphatic form of ne. Subsequently it became unstressed and supplanted ne altogether. This is an example of Jespersen’s Cycle.

All these things combined led to a new negative form of willan, wynnot. The past forms of willan began with wold-, which is where we get would. Under the influence of these forms and the related verb wol, wynnot became wonnot by the late 1500s.

Finally, the modern form won’t emerged by the 1660s as a result of reducing the final vowel in wonnot. It appears to be the first word so contracted; most of the other -n’t contractions we use today (can’t, couldn’t, shouldn’t, &c.) arose in the 1700s, modelled after won’t. In modern English, cannot is the only uncontracted -not compound that survives.

As for the other contractions such as -’ll and -’ve, their history is just as long, though perhaps slightly less convoluted. But that’s a story for a different question. ;)

Also, remember that spelling in Old English was less standardised than in modern English. There were often several equally valid ways to spell the same word, especially when you took different accents and dialects into account. So sometimes it’s difficult to get a good historical account of pronunciation and usage changes. Still, as far as I can tell, this is basically how it went down.

Source: The Online Etymology Dictionary.

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.