A light wave has a frequency.  But a photon, travelling at the speed of light, does not experience the passage of time.  How can it exhibit a frequency then?

Photons don't have frequencies, at least not in the sense implied by the question. The following is basic overview of wave mechanics in the Minkowski spacetime; essentially Landau and Lifschitz Vol. II, ch. 7 adapted into a picture book for the well-informed non-scientist. Though lengthy it is not meant to be exhaustive, only to convey enough for the reader to  understand the answer to the question.What Exactly is a Photon?The ontology of a photon is not firmly established, nor do we fully understand every implication of the Minkowski spacetime. It should suffice for the discussion here that a photon is the quantum of electromagnetic energy/momentum exchanged between emitter and absorber.The Null SurfaceIt would be impossible to describe wave mechanics in spacetime without understanding the causal structure of spacetime. It is prerequisite to know that massive objects are timelike (live inside the lightcone), massless objects are lightlike (live on the lightcone), and no real particles are spacelike (live outside the lightcone). If you don't, not a big deal, I have handy these 2 references and the picture below for reference: For simple see: for all the details see: https://en.wikipedia.org/wiki/Light_cone Affine Parametrization: The Photon gets its Color Imagine suddenly appearing in the woods, standing on an unmarked path. You walk along the path, placing evenly spaced markers alongside the path to indicate distance, time at each marker, or some other addition information. The line of markers tangent to your path functions like an http://planetmath.org/affineparameter. In relativity we can do this easily for a timelike path and use the traveler's wristwatch time,  called their proper time, which makes for a perfect affine parameter.It's different for a photon: The entire path through the woods, the forest itself, collapses to a single point. Using proper time is completely meaningless in this context. However, in our spacetime we do see a path of the light ray that tracks through our spacetime and it is still possible to define an affine parameter so long is it meets two criteria: The affine parameter must satisfy the geodesic equation The tangent vector to the photon worldline must have zero length With a choice of affine parameter that meets these criteria λa=τω+c\lambda_a=\frac{\tau}{\omega}+c where λ\lambda is the affine parameter (no relation to wavelength), Ï„\tau is the proper time of the local observer, ω\omega is the frequency, and c is an arbitrary constant.With this, it is possible to construct an appropriate null tangent vector called the wave 4-vector: If it is not clear why this is zero we can use c=λνc= \lambda \nu The Wave 4-Vector The wave 4-vector was constructed by starting with the energy-momentum 4-vector. Then a direct substitution is made using the relationship between a photon's energy and momentum and its wave properties. The energy-momentum 4-vector is just the wave 4-vector scaled by h-bar.Isn't this Just a Contrivance?? It looks that way... we have a null surface, with a null tangent, and even the normal vectors to null surfaces are themselves null, so it looks like we can't meaningfully connect our own timelike reality to the null realm of the photon. So do we just take our photon's "path through our space" and mark it with an arbitrary tangent line set equal to a classical wave vector, with the only reason being because we know the physics has to somehow end up that way? Actually, the physics works out quite nicely. To understand how we need to dig a little deeper. The Eikonal Surface: What the Null Surface Encodes First, let's look at the oscillator and the E-field component of the resultant electromagnetic plane wave. Image from http://philschatz.com/physics-book/contents/m42440.html The emitter begins at T =0 with a maximum value of the electric field and as the phase of the emitter changes the variations in the electric field are shown to propagate to the right.The emission of the wave and its absorption by say, your eye, is a single event on a null surface connecting the two events (two events as recorded by the timelike observer). Each of the vertical arrows indicates a particular phase of the wave and as each phase is an event in spacetime, each has its own light cone.  What the light cone represents for a wave is a surface of constant phase, called an eikonal surface, and each surface records the phase of the emitter at some time "t".This can be depicted in Minkowski spacetime as shown below. The t-axis defines the emitter's worldline and the line parallel through "A" is the worldline of the observer at rest with respect to the emitter. The five light cones represent the first five phase points, the blue lines running right to left, in (d) of the diagram above.The segment OB is null separated and exists as a single event. The segment AB is the timelike separation between emission and detection as seen by the observer. A Story To Go With the Picture At point "O" is your television, and you're sitting in your chair at point "A." The worldline along "t" is some atom or some pixel on your TV. Your are both at rest w/respect to each other, although moving through spacetime at the speed of light. This is trivially shown by the tangent to either worldline: You turn on the TV which excites an atom/pixel/oscillator. You see nothing until your motion along your worldline carries you through the first null surface at "B", which encodes the phase of the emitter at "O." As you tear through the eikonal surfaces (light cones), each with its own phase, you experience a "wave" with a frequency set by oscillator and experienced as the temporal spacing of the light cones. If the observer moves away from the emitter and accelerates after the light, the observer asymptotically approaches the eikonal surface, and a wave with a constant phase is effectively a wave with an infinite wavelength. In other words, chased photons are redshifted. Moving towards the emitter, the observer sees blueshifted light. It should be noted that spacetime is a smooth manifold and so there is a continuum of lightcones that form a null subspace.In summary: It would be wrong to imagine a photon as a little flashing particle. Perhaps it is easiest to imagine the photon not existing at all and consider only the emitter and absorber joined by a null surface. The frequency we observe is then just a byproduct of our simultaneous motion through spacetime along with that of the emitter.

As a general rule its not good idea to try an examine the philosophy of unphysical objects, you get into really murky territory. The light-like reference frame is a limit of physical frames as you approach c, but it in and of itself isn't really a physical frame, since you can't accelerate into it with finite energy. Its like a pure momentum or position eigenstate in quantum mechanics, useful for math but you shouldn't spend too much time thinking about it (if you don't know any QM then I apologize for confusing you). In addition, you're making another mistake when you talk about the photon's frequency relative to itself. The frequency of a wave is related to the change of phase in time. The phase is given as e(iwt)e^{(iwt)} and you can see that the derivative w.r.t time is proportional to ww, the frequency. In both classical and quantum mechanics there is no such thing as absolute phase, only the relative phase between too different objects or between and object and itself at another time or place is physically relevant. So in short, a) don't mentally accelerate into a light-like frame and b) don't create an absolute phase.

I believe the main issue here is with the use of the photon as a conceptual model for propagating light. The idea of the photon oscillating up and down in space (the frequency, which does indeed require time to occur) seems counter intuitive because we conceptualize the energy of the wave as quantified into discrete particles (which must travel through the structure of the waveform in order to create the field patterns light exhibits). If however, we view the light wave as a unified continuous field, then it is much easier to understand how this phenomenon occurs. When the field is considered as an indivisible whole, the necessity for photon displacement no longer exists, as the energy is dispersed in an extended continuous fashion throughout the wave.

Fly along the ocean just above a wave train and it might look like it's completely static relative to you. And yet someone on the beach watching the same waves go by sees up-and-down motion with a definite frequency.

Assuming photons does not experience the passage of time (which is true AFAIK), there is no reason why it should not exhibit a frequency. Think about a rock, which theoretically doesn't exhibit time (although it does). Let's say you decide to break the rock 1 million years after it was created. It's state changed, although it wasn't aware of it. The same goes for a photon. There is no contradiction whatsoever. Objects which do not experience the passage of time doesn't necessarily mean that they do not deform over that period of time.

Well the photon doesn't experience time but we see it experiencing it, and therefore we interpret its energy as frequency. But frequency is just an interpretation of us, time-like objects. According to the photon, ie in its frame, energy is just energy.

Time only appears non-existent to the photon. To massive particles in the universe frequency is a property of the photon which dictates how, or not, it interacts with those particles. To use Mr Reiter's example of a rock, a rock has a mass but has no awareness of that mass. You on the other hand interact with the rock and can determine its mass etc. Not a brilliant analogy I know.

This is a wrong notion. Time does not get slower for the object. The object itself does not experience any time dilation or length contraction. When one observer in an inertial frame observes the object to be in relative motion w.r.t. itself, he or she observes that the object is contracting lengthwise (in the direction of motion), and that the time of the object is running slow. In the same fashion the object also observes the observer in relative motion and has the same observations about the observer that the observer has for the object. But the object finds its own time and length to be unaffected or unchanged by the motion, and so does the observer.