If we have only reactive elements (i.e., none resistive) in our load, where does the power delivered to it go if it's not dissipated?

If you have a purely reactive load (purely imaginary impedance), the phase shift $\phi$ between current and voltage will be $\pm \pi/2$ and average power dissipated (proportional to $\cos \phi$) will be zero. This corresponds to a situation where energy is sloshing around between capacitors and/or inductors and never gets dissipated. (In practice, of course, nothing really has zero resistance, so energy will always be dissipated, even if slowly. In superconductors, energy dissipation is really tiny though, and energy can be stored for a very long time!)