Noah: The sentence "there is a real not in $L$" is $\Sigma^1_3$: To say that $x\notin L$ means that for every $y$, if $y$ codes a model of the form $L_\alpha$, then $x$ is not in this model; but to say that $y$ codes an $L_\alpha$ (for a sufficiently "closed" $\alpha$) means that $y$ codes a structure $(M,E)$ (this is arithmetic) that satisfies, say, $KP+V=L$ (you can do this in, say, a $\Delta^1_1$ way); and you need to express that this structure is well-founded, for which you just have to say that no real codes a decreasing sequence through its ordinals.

Counting quantifiers, this comes out $\Sigma^1_3$. It is false in $L$ and true after adding a Cohen real.

On the other hand, if $\phi$ is $\Pi^1_3$ and holds in an outer model of $V$, then it holds in $V$: The sentence says that every real (in the extension) has an absolute property, so in particular every real in $V$ has that property, in $V$.

This, however, is not "the limit of absoluteness," as per your title, only of Shoenfield's absoluteness. Large cardinals in the universe grant you much stronger absoluteness properties between $V$ and its forcing extensions. Even at the level of (boldface) $\Sigma^1_3$ generic absoluteness, this involves sharps. A nice proof is sketched in my paper with Ralf Schindler, "Projective well-orderings of the reals" Archive for Mathematical Logic 45 (7) (2006), 783-793, available at my page.

Beyond this level but still looking at projective statements, strong cardinals are involved, by a nice result of Woodin. You can read the details in John Steel's "the derived model theorem," which also gives you a nice introduction to some of the topics that come up when studying absoluteness, such as universally Baire representations of sets. The paper is available at John's page.

Even beyond this level something can be said. Now you require Woodin cardinals, and absoluteness is tied up with determinacy. In fact, the statement that "no set forcing can change the theory of $L({\mathbb R})$, even allowing reals as parameters," is equivalent to saying that determinacy holds in $L({\mathbb R})$ in any set forcing extension. This in turns is equivalent, for example, to the statement that the mouse $M_\omega^\sharp$ exists and is *fully iterable*. $M_\omega^\sharp$ is a fine structural object (a generalization of a level of $L$ or of $L[\mu]$) with $\omega$ Woodin cardinals. Full iterability is a technical condition, but it ensures that in any forcing extension, $L({\mathbb R})$ is a *derived* model, a requirement from which determinacy follows.

And we can even continue beyond $L({\mathbb R})$ for a bit. But I'll stop here.