Objects possess potential and kinetic energies. The potential energy depends on the position of the object, while the kinetic energy is dependent on its velocity.
My question: are there other types of energy (corresponding to, say, acceleration, jerk, etc.)? Why are there only two fundamental types of energy?
Higher derivatives of position don't have their own corresponding types of energy, because they're not independent quantities. (See this or this or any of several similar questions.) $F = ma$ relates the second derivative ($a$) to the position itself ($F(x)$), so if there were a type of energy that depended on acceleration, you could just re-express it as a potential energy.
To make a slightly more general argument, energy is a state variable: a system's energy depends only on its state. In classical mechanics, for a single particle, the state is determined completely by the position and velocity, so the energy can only depend on those. In quantum mechanics, the state is determined by the wavefunction, so the energy can only depend on the wavefunction. In thermodynamics, the energy can only depend on temperature, pressure, and particle number. And so on.
My question: are there other types of energy (corresponding to, say, acceleration, jerk, etc.)?
There are other types of energy, but not the way you're suggesting. Non-gravitational F=ma acceleration is the result of a change of energy, and a jerk is just a rapid acceleration.
Why are there only two fundamental types of energy?
I wouldn't say there are only two fundamental types of energy, in that we talk about things like photon energy, neutrino energy, gravitational field energy, dark energy, and so on. But I would say this: potential energy isn't fundamentally different to kinetic energy. It's just hidden kinetic energy.
I suppose I'd better try to explain why I say that. OK, take a look at Compton scattering:
Image courtesy of Rod Nave's hyperphysics
In Compton scattering, some of the photon's E=hc/λ wave energy is converted into electron kinetic energy. If you performed another Compton scatter using the scattered photon, and another and another, then in the limit you remove all of the photon kinetic energy, and there's no wave left. The photon has then been entirely converted into electron kinetic energy, which makes perfect sense, because light is just kinetic energy. But in pair production you can convert the photon into an electron and a positron, so you can say the electron is quite literally made from kinetic energy. Then when you lift the electron up, you do work on it. You exert a force on it for a distance. You add energy to it. We call this energy potential energy, but the electron is still made of kinetic energy. You know this because when you annihilate it with a positron you get two photons, which are just kinetic energy.
However if you don't annihilate it, if you drop it instead, we say that potential energy is converted into kinetic energy. But IMHO what's really happening is that internal kinetic energy is being converted into external kinetic energy, and as a result, the electron falls down. Then when you dissipate that kinetic energy as radiation, you're left with a mass deficit, because there's less kinetic energy in there. IMHO it's all much easier if you've been taught about the wave nature of matter, because then you can really appreciate what E=mc² is all about.
