Why do Rydberg atoms have electrons that can transition above the first ionization energy?
A useful concept in Rydberg atoms is the notion of a channel. This concept is borrowed from scattering theory and describes all solutions of the Schrodinger equation for the scattering of the Rydberg electron of specified angular momentum with the ionic core in a certain energetic state. The bound Rydberg states correspond to scattering at negative energy and the scattering at positive energy corresponds to the ion-electron continuum of the channel.
If the ion has multiple electrons, it can have several excited states. Each of these excited states can form a channel. The energy of a Rydberg electron is given by
$E=E_\text{IP} + E' - \frac{hcR}{(n-\delta_\ell)^2}$
Where $E_\text{IP}$ is the first ionization energy, $E'$ is the excitation energy of the ionic core, $R$ is the mass-corrected Rydberg constant, $n$ is the principal quantum number and $\delta_\ell$ is the quantum defect for a Rydberg electron with orbital angular momentum $\ell$.
Each excited state of the ionic core thus gives a new Rydberg series with each own set of principal quantum numbers. So, Rydberg states do exist above the first ionization energy, but their principal quantum number is finite; it just belongs to a different channel.

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