When is energy discrete/quantized for a potential well? Specifically, my question is:
Should one expect energy quantization for a particle in the following potential well?

More generally, how can one tell whether or not energy should be  discrete/quantized just by looking at the potential function?
 A: The rule of thumb for me is how would the solutions of the Schrodinger equation  look for this potential. When the energy of the electron is above the potential well the solutions can be continuous ( although there can be discrete resonances with an energy width). Below the lip of the potential, as for example in the hydrogen atom, the solutions will be discrete, quantized. 
If the walls of the potential go to infinity, as in your example,  then all the solutions will be quantized as in this example of the particle in a box .
In the limited in energy potential ( not going to infinity) the energy levels with large energy quantum numbers are very dense and resemble a continuum, although they can be labeled with a discrete number, as for example in the hydrogen atom. 

Above the energy of the well are the continuum solutions, not discrete.
A: Well, if we start from Bohr's explanation to the stability of the atom, we remember that he said that the obit of the electron should make a standaing wave around the nucleus. Quantum mechanics came up, then, and showed that the situation of the standing wave is true always when you have a potential that isn't time dependent.
So, in any potential you have, you should have standing waves, and standing waves create your quantization condition. That's the answer in the simplest form.
A: The regions with $U > E$ are classically forbidden - a particle cannot get there, it is reflected (stopped and sent back). In QM it is the same except the wave function can penetrate slightly into forbidden regions. As far as the wave is reflected, it exists within the well. Such a wave can be represented as a sum of standing waves with some time-dependent coefficients. A standing wave has a certain energy $E_n$ because the wave with $E_n <E<E_{n+1}$ interferes differently (it is not standing) and its energy is in fact uncertain.
A: One way of getting to the answer, which is probably not going to cut it for Jonathan, is to use the WKB approximation to get at the wave-function. This involves matching coefficients inside and outside the forbidden regions of the potential, and this matching can only be possible for certain values of $E$. This leads to the Born-Sommerfield quantization rules. Roughly, the action of the particle moving from one end of the classically allowed region to the other is a multiple of $2\pi n \hbar$.
A: A summarised derivation for the 1 D triangular potential well can be found here. Interesting graphs of the wave functions $\psi_n(x)$ are provided.
Also treated at [this *.pdf] (p.5, section 1.2.8.2)2.
