Is it theoretically possible to have a photon of any energy/wavelength? I have a vague memory in my mind of something like there being a minimum energy level for a photon and then the possible energy levels jump up in discrete but ever decreasing amounts, so that near e.g. the visible spectrum photon energies are for practical purposes continuous.
The photon is a quantum mechanical particle in the standard model of particle phyisics. Classical electromagnetic radiation emerges from a confluence of innumerable photons.
The photon is characterized by its zero mass and its energy which is equal to h*nu, where nu is the frequency of the emergent classical wave.
What you are referring to are photons from transitions in energy levels of atomic or molecular or lattice bound states. There the photon comes in quanta, but these transitions are different for different atoms/molecules/latices . Once emitted they can be redshifted or blue shifted depending on the emitting source's velocity, so there are spectra , but no real constraints on the possible energy carried by a photon. So yes, even though the photons may be coming from individual spectra, the frequency can seem continuous because of the great multiplicity of atomic/molecular/lattice sources.
In addition, charged elementary particles emit photons when accelerating or decelerating. In an antenna for example, the accelerated and decelerated electrons emit an electromagnetic wave: this is the confluence of individual photons emitted by individual electrons within the antenna, and the frequency is variable and constrained only by the possible dimensions of the antenna.
In conclusion photons are only constrained to have zero mass, but can have any energy, i.e. any nu. This is true for massive particles too, which are constrained to have their fixed mass but can have any energy.
There is a minimum energy level $h \nu$ (corresponding to a single photon) for light of a given frequency $\nu$. But there is no absolutely minimum amount of energy for light, because you can have a photon with arbitrarily low frequency $\nu$ (ignoring the possibility that the universe is finite in size).
The introduction of photons comes from the empirical fact that energy transfer happens in portions. Einsteins famous explanation of the photoelectric effect had underlined the theoretical work from Planck about the quantization of the black body radiation. Photons are indivisible units from their emission until their absorption.
The emission of photons is associated with energy losses. Every photon carries energy. Furthermore through double slit experiments the spectrum of light was associated with wavelengths and using the speed of light with frequencies. From thermic sources (non-modulated electromagnetic radiation) one get radiation from gamma, X-ray, ultraviolett, visible to infrared. The named radiations are less and less energetic per photon. The minimum level for photons lay in the range of infrared respectively microwaves.
Now somebody says that radio waves have lower frequencies, longer wavelengths and by this radio waves a lower energy level. This is nonsense. Radio sources are modulated radiation where electrons periodically get accelerated and by this emit EM radiation in the range from IR to X-rays. Please consider that for the same radio frequency the source can have very different energy content which depends from the wave generator.
Providing spectroscopic experiments it was find out that atoms and molecules emit photons of some wavelengths and by this always in the same relation of the emitted wavelengths. The shift in the energies of the emitted wavelengths depends from the energetic level of the emitting body.
As long it was not measured any quantum behavior of the energy emission. Changing the energy content of the emitting body the spectral lines changes continuously. This does not mean that there is no quantisation of possible photon states, but it is not observed until now. A mathematics based not on statistical properties but on the existence of smaller particles as constituents of photons would be a task for the future.