So, what's the joke in number $9$?
$9$. You understand the following joke: $\forall \forall \exists \exists$
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For a joke to convey its point, it has to allude to some nearby cultural association that many people in the audience will understand. By far the most recognizable instance of $\forall ... \exists$ statements, or any statement with more than one type of quantifier, is the epsilon-delta formalism in calculus, and the misery that is said to induce in students. It is a rite of passage in mathematics education and therefore a sign on the road to being a mathematician.
I think that the joke is alluding to the word pattern of the epsilon-delta incantations and, to some extent, its stereotypically associated emotional reaction, as a synecdoche for mathematical language, mathematicians and their habits, and higher mathematics. A student who has suffered through the calculus classes knows that as soon as the words "for every [epsilon]" appear, "there exists [delta]" will follow, with all the complications attending that.
Hence, "for every $\forall$, there is a $\exists$". There is some implied commiseration in the joke, a reference to something notorious that the audience has been through, equivalent to
"for every $x$ there is a $y$" (school algebra)
"for every superscript there is a subscript" (tensors)
"for every grad there is a curl" (multivariable calculus)
There is also the idea that mathematicians are those who have passed through that perplexing rite and (in addition to being able to commiserate) are the ones who are fluent and comfortable with it, who can understand the joke.
My guess is the joke was just not executed that well; the author of spikedmath had something in mind, but we have no idea what it was. However, here are some possibilities I can think of:
Very often in mathematical statements, $\forall$ and $\exists$ quantifiers alternate. $(\forall \epsilon > 0)(\exists \delta > 0) (\forall x) \ldots$, for example. It is common to transform any formula into one in prenex normal form, and sometimes when we do we additionally assume that the quantifiers alternate, for convenience. So $\forall \forall \exists \exists$ could just be the joke that the quantifiers don't alternate. It's funny in the same way as "let $\epsilon < 0$": it's not the way they are usually used.
You gave a link in your comment in which someone says the joke is "for every forall symbol, there is a there exists symbol nearby".
The joke could be simply that it is syntactic nonsense. We expect to see a variable after the $\forall$ symbol, but we do not. Instead, $\forall \forall \exists \exists$ seems to be quantifying over another forall symbol and then over another exists symbol. So it reads like syntactic nonsense that seems like it should make sense but doesn't, and may strike someone as funny in this way.
I read $$\forall\forall\exists\exists$$ as
whenever there is a statement with $\forall x: A(x)$ there is also a statement with $\exists x: A(x)$.
Mathematically this is a triviality, but its amusing that we formulate different levels of semantics using the same symbols. It's some funny kind of smearing different levels of information by means of one syntax.
What struck me (personally) as droll was:
It can be construed as redundant, in that two "for every" clauses in succession can be replaced by a single "for every", and similarly for two successive "there exists" clauses.
It looks like the cry of a cartoon character falling upside down from a great height.
After reading the existing responses here, I lean toward the interpretation "whenever you see a $\forall$, a $\exists$ lurks nearby", a wry comment on epsilontics.