But what is this actually trying to say?
This ($\exists x \exists y (x\geq y $)) is actually just saying that there is a real number $x$ such that there is a real number $y$ such that $x$ is greater than or equal to $y$.
Note that technically the $y$ could (will) depend on the $x$. So the specific choice of $y$ might depend on $x$. This makes sense because if $x =3$, then you could pick $y = 2$, but if $x =1$, then you can't pick $2$, but you obviously need a number less than or equal to $1$, say $y = 0$.
Now one might also think of the statement as: there are two real numbers such that one is greater than or equal to the other. Or: There are two real numbers such that one is not greater than the other.
If you want to say that
a number is as large as itself
Then you are saying that given an $x$, this number is "as large as it self". The "as large as it self" might be a bit confusing to some. But one might interpret this to mean that $x\geq x$. So one could write the statement like: $\forall x (x\geq x)$. I would disagree with the saying that the statement is $\exists x (x\geq x)$ because this statement is just saying that there is just some $x$ such that $x$ is "as large as it self". It doesn't say that any real numbers are "as large as it self". The "a number" part I would interpret as "for all numbers". Also note that in this statement (a number is as large as itself) we are only talking about one number.