What's an example of a device that uses both a capacitor and inductor

a radio

AC filters, which are used in audio and RF electronics: Hgh-pass, low-pass, band-pass and band-stop (or "notch") filters.

Analog oscillators (signal generators) found in older electronics.

Audio power amplifiers.

Stun guns.

Etc...

BALLAST -- inductor to limit the current and generate counter EMF to start the light. It also contains a capacitor to correct its power factor.

I think power supplie boards have both, to filter out the ac electricity.

A fluorescent light