Question

What happens at the ends of the electromagnetic spectrum? Do the wavelengths continue to get shorter and longer or is there some ultimate limit?

The electromagnetic spectrum has no theoretical limit at either end. You can create as long a wavelength (ie. as low a frequency) as you want simply be moving an electric charge back and forth in as long a period of time as you choose. Given the proportionality of energy and frequency, long wavelengths such as this represent very low energies.

At the other end of the spectrum, it takes higher energies to create higher frequencies (shorter wavelengths). That means you are limited only by the energy available. If all the mass/energy in the Universe is considered a 'limit', then that would be the only real theoretical limit to the maximum frequency attainable.
Answered by: Paul Walorski, B.A. Physics, Part-time Physics Instructor

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Science Quote

'Physicists like to think that all you have to do is say, these are the conditions, now what happens next?'

Richard Phillips Feynman
(1918-1988)