I understand that a water heater element is basically just
adding energy to the mass of water in the container. So does
that mean that the heater element only has to keep adding energy
while the water around it becomes hotter than the element
itself? Or does the element warm up together with the water
until both reach 100C? The water will not warm up, if the heater
has the same temperature as the water. It needs to be hotter
than the current water temperature. You can not "add" thermal
energy to the water by connecting an object that is in thermal
equilibrium with the water. In practice the element will get
very hot without any water cooling it and it will eventually
switch off to prevent any damage. If you add water this process
is slowed down a lot, as the water has a high heat capacity but
until the heater switches off it will always be warmer than the
water. The amount of heat flowing per time depends on the
temperature difference as stated by Fourier's law: $$\partial Q
\propto \Delta T$$