Can a photon have any amount of energy? Since $hc/\mathrm{wavelength}=$ Energy of the photon, and the wavelength can be anything, $0.0015465$ m for example, does that mean that the photon energy can be anything?
I heard that photons can only carry discrete amounts of energy .
 A: Yes, a photon can have any energy.
(Unless you get into crazypants speculation about discrete spacetime, quantum gravity, Planck lengths, etc. But I don’t think that’s what you’re asking about.)
Note that photon energy is observer-dependent, thanks to the relativistic Doppler effect.  So if you’re sending me photons at an energy I dislike, I can adjust their energy in my reference frame by walking towards or away from you.
You can’t absorb a fraction of a photon’s energy, which is what people generally mean when they talk about “discrete amounts of energy.”  The energy comes in lumps (or “quanta” if you’re sophisticated), and you can absorb the lump or not. But the lumps can have any size.
A: Both things that you said are correct: a single photon can have any amount of energy, but that amount of energy is discrete for a fixed wavelength, in the sense that you will find only integer multiples of that energy $hc/\lambda$ at that given wavelength, and the photon can only ever deliver energy $hc/\lambda$, not some fraction thereof.
A: No, a photon cannot have any amount of energy.
The proof is simple. Photons are produced exclusively by their emission from subatomic particles. In an annihilation process of proton and anti-proton at least 2 photons are produced. The higher the kinetic energy with which these two particles collide, the higher the chance that new elementary and composite particles are created besides photons. The theoretical possibility to get exactly two photons with the energy of the particles accelerated against c is not reached in the experiment.
Now people like to refer to the oscillating dipole, i.e. an antenna. Their EM radiation can actually be scaled upwards. However, if one goes into detail, one inevitably comes to the conclusion that the synchronous acceleration of the electrons on the antenna rod to the individual emission of polarized photons comes from each individual accelerated electron. Thus, the EM wave consists of photons whose individual energy content is determined by the acceleration rate of the individual electrons.
A higher energy EM wave is achieved solely by increasing the number of accelerated electrons and either increasing the frequency of acceleration or lengthening the antenna rod. But that's another story, where radio engineers struggle with the energy transition between the antenna current and the conversion to a photon current in vacuum (or air, as the case may be).
