What force decays an oscillating charge? Consider this scenario: We have a charge on a spring. As it oscillates, it loses energy to radiation.  
What force decays the oscillation? As far as I can see, the only forces in the scenario are electromagnetic forces, and the spring.
The spring force is ruled out because it's energy conserving and doesn't decrease the oscillation.
Electromagnetic forces are ruled out because there are no other charge in the scenario to exert a force on our charge.  
What am I missing? Is this a force not described by Maxwell's equations? Does the charge exert a force on itself? Something else entirely?
 A: The technical term for this force is the radiation reaction force and it is electromagnetic in nature. 
Maxwell's equations do not describe this phenomenon simply because they're not meant to; it's like asking the heat equation to describe the chemical reactions that happen in a fire. Maxwell's equations describe the electric and magnetic fields generated by a configuration of charges and currents, and to form a complete description of nature they need to be augmented with the reverse connection: how the fields act on the charges and currents.
This second half of the description is usually done with the Lorentz force, $\mathbf F=q(\mathbf E+\mathbf v\times\mathbf B)$. This works well for continuous charge distributions (where self-interaction effects vanish as each individual charge is infinitesimal) and for point charges where you can identify and subtract the own field of the particle. For an oscillating, radiating point charge you can do neither of these and you need to extend the Lorentz force to a more general version. (A better way to see this is that you need to undo the approximations which gave you the Lorentz force.)
This process is slightly tricky and even within classical electrodynamics there are nooks and crannies for which our explanations are not completely satisfactory. The best first stab, however, is the Abraham-Lorentz force,
$$
\mathbf F_\text{AL}=\frac{q^2}{6\pi\varepsilon_0 c^3}\frac{\mathrm d\mathbf a}{\mathrm dt}.
$$
To give an example of the difficulties presented by this force, note that the equation of motion is now third order in time. There are multiple ways to derive the Abraham-Lorentz force, but ultimately this needs to be embedded in a bigger framework - if nothing else, to account for the third-orderness of the equation of motion. If you want to learn further, Wikipedia and Duck Duck Go are good places to start - or ask a more detailed question on this site.
