You already have plenty of references, but let me give you another one.

The *standard quadratic transformation* is the birational map $\sigma: \mathbf{P}^2 \dashrightarrow \mathbf{P}^2$ defined by

$$ \sigma: [x_0, x_1, x_2] \rightarrow [x_1x_2, x_0x_2, x_0x_1].$$

This maps fails to be defined at the three coordinate points; in the classical terminology, one says it is *based* at those points. Blowing up the plane at the three coordinate points, $\sigma$ then lifts to an automorphism of the blowup; moreover, conjugating $\sigma$ by elements of $PGL(3)$, one gets an automorphism of the blowup of $\mathbf{P}^2$ at any triple of (non-collinear) points.

The idea is then to compose maps of this kind (based at different triples of points) to produce $(-1)$ curves on the blowup of $\mathbf{P}^2$ whose degree downstairs is arbitrarily large. That is, start with any $(-1)$-curve $C$ you like, for example the proper transform of the line through two of your points. Then apply a sequence of Cremona transformations such that the successive images of $C$ are curves whose degree gets larger at every step.

But how do we know such a sequence exists? This can be translated into a problem about root systems and Weyl groups. The key point turns out to be that for the blowup of $r \leq 8$ points the root system has finite Weyl group, whereas for $r=9$ the group is $W\left(\tilde{E}_8\right)$, which is infinite.

For details about the last paragraph, the source I know is the paper "Weyl Groups and Cremona Transformations", by Dolgachev.