To me, the other elemntary particles are equally mysterious. This is because of their non-intuitive nature.
Naive ontology of the world
We humans grow up in a world with objects. These objects have mass and volume, they have discreet boundaries. Our brains are wont to consider the objects as separate things, and each thing can, for instance, be picked up and otherwise inspected by the senses. Things are accordingly denoted by ordinary nouns.
Then there are fluids, which are almost like things but not quite. They do have mass and volume, and they can be interacted with through the senses (although to realize that air is not vacuum by noting air resistance is a non-trivial leap of insight), but they are not separate. They merge with each other and can be arbitrarily divided, unlike things, so you can never put your hands around exactly what a fluid is. Accordingly, we know them by uncountable nouns, and understand that ultimately it makes sense to think of parts of a single universal fluid. The universal fluid becomes the only thing, since other subdivisions are moot, except in the special case of droplets or quantities sequestered in vessels, where the thingness is being artificially enforced (ie. you can pretend the coffee and the milk are separate things so long as they are in their separate cups, but the moment you let them come in contact this pretense falls apart).
Everything besides these we think of as phenomena. For instance, fire is something that happens: It is not a thing you can pick up and manipulate (you can only pick up the fuel, and the fire clings weirdly to it), nor is it something that you can trap in containers and subdivide or combine (though the fuel may be either a thing or a fluid, and it can be operated on, with fire sometimes coming along for the ride). Likewise for sound, light, temperature and similar concepts.
Of course we now know fire is just plasma, ie. a fluid that "spoils" into something else very quickly, and that it is in fact possible to trap it with the right exotic vessel. Thus the imagination may be coerced into accepting fire as a fluid, although in everyday life it still appears as a phenomenon, thus it is not a truly intuitive thing.
The theory of the atom
When the atom theory came about, I'm certain the Greeks at some point expected the "atoms" to have some mass, shape and size, ie. to be things. Thus, a neat conceptual trick is employed: To the careless eye, sand seems like a fluid, since quantities of it appear to freely merge and split (note that sand is uncountable). But on closer inspection, the sand is just a bunch of tiny objects, which themselves are clearly things.
The realization then came that the world would make sense if everything was a kind of sand made up of many tiny particles, which are themselves things. This is nice because it unites fluids and things: The fluid is only an apparent class, and deep down they're all things - which is nice. Things are very intuitive for us humans, and it can be easier to reason about a pile of millions of tiny things than a single bit of fluid because of how weird fluids are. Luckily, it turned out that both fluids and things were made up of particles, and these particles do seem like bona fide things.
Here the caveats begin. Molecules are not quite Newtonian solids. They behave almost like them: For instance, they can have mass and volume. Almost all of them can be broken apart, but if you consider the very rigid rule of breaking up a molecule vs. breaking a rock, it starts looking funny already. They do have a boundary and bump off of each other... But watch out that you don't bump them too hard, or they weirdly merge together (unlike rocks). But the worst part is the boundary, which is only a fake boundary: The Van der Waals radius is not a binary "can/cannot pass" delimiter, but is a consequence of a continuous force equation. It isn't really that much harder to be slightly inside the molecule than slightly outside. Compare being slightly inside of a rock -- impossible.
As an aside, I think it's interesting that the Greeks came up with a theory of the atom rather than a theory of the fluid, where all solid objects are in fact fluids in some temporary state of rigidity. The various theories of the elements come to mind, but they don't make the right physical observations: One could observe discrete pieces of iron can be melted and seemlessly combined, and then conclude that surely it must be possible to melt any thing, therefore there are only apparent things, and everything is essentially a fluid. Perhaps it is because this theory of fluids makes the world more confusing, not less.
Molecules, as it turned out, where simply small aggregations of things -- surely when you break up a thing the result must be smaller things? We soon found out about atoms, and then the parts of the atom. This is where we stop, since to my knowledge, none of the elemntary particles are known to be divisible into further constituents. Photons are one such elemntary particle.
The pretense of a thing may be maintained for atoms and molecules through devices like the Van der Waals volume. For elementary particles, this pretense is hopeless. As has been famously shown time and time again, not only do elementary particles not have volume, they blatantly don't have volume: If they did, physics just clearly doesn't work, and you get things like "surfaces" of electrons spinning faster than the speed of light. The observation was then made that the world would make sense if only these particles were points.
Of course, nobody actually knows what a point mass is. Nobody has ever seen such a thing (well, except that which we wish to christen a point mass in the first place). Its implications seem bizarre: For instance, its density is infinite, and in theory the entire universe could be squeezed to a single point. Had particles been things, such insanities would be comfortably precluded: Rocks cannot be squeezed arbitrarily, not even with infinite force.
Not being squeezable, by the way, is another property of things. Even soft things like sponges turn out to just have pockets of air in them. Once the holes are all squeezed out, a thing cannot be compressed further: Liquids politely pay lip service to this principle, although you can tell (eg. by the water in a syringe experiment) that their hearts really aren't in it, and gases just couldn't care less -- another way in which fluids are strange and unlike things. Or at least, to a naive observer without access to the extreme energies required by our modern physical experiments.
The subatomic level is where intuition completely breaks down. You can create analogies, such as to strings and pots of water, but you can never really imagine what a particle is like in terms of what objects from everyday life. The Universe has played a very cruel trick on us, in that it is one way, but it is such that at the macro level at which we necessarily started to understand it, it is entirely in an other way, with no semblence of the one way to be seen. We are then doomed to grow up expecting and getting used to the other way, only to take Physics 201 in college and find out everything we know is an illusion and no intuition is possible for the true nature of the world. Intuition, indeed, is an comprehension based on experience: Who can experience the subatomic? At best we may experience experimental apparatus.
The top-down approach to understanding the Universe fails, and it fails precisely at the subatomic level.
The bottom up version
One can debate the true meaning of intuition, but I think some sanity can be restored by instead starting over and setting everything right. We can forget all the naive baggage about things and fluids, wipe the slate clean, and start with the fundamental truth that in the world, there are particles. Particles have momentum, they are points, they interact with each other and the vacuum in certain ways described by quantum mechanics. They are elementary, and not made of any smaller units. Photons, then, are one such particle, with specific properties described elsewhere (I won't repeat them, since you explicitly said in your question that these descriptions are not what you want).
When truly large numbers of the particles act together, at the macro level some bizarre phenomena come about, such as "volumes" and "state transitions". You can't really get an intuition for these bizarrities from our knowledge of particles. But logically, ie. if you follow the math, you know it is a simple and straightforward consequence, albeit non-intuitive.
Unfortunately, this bottom up intuition is not very useful. All of our daily lives concern macro phenomena. A lot of the interesting things in the universe (basically, all the disciplines other than subatomic physics) are macro scale. One expects that after learning physics, the world will become easier to understand -- but learning the bottom up intuition only makes everything harder. I suspect even subatomic physics is not made much easier, since all the real work is done with math, not intuition.
So, in conclusion, the question cannot be answered satisfactorily. There are two ways of understanding a question like "what is an X":
- "Tell me the salient properties of X": For the photon, there is no shortage of various texts, and even on this site there are perfectly serviceable answers that you have found lacking.
- "Help me intuitively understand X": As I said, no intuition is possible without tearing down all the intuition you have built up over your life. If you do tear it down, what intuition may be gotten is unsatisfying, and only serves to give you a headache.
But that said, a photon is an elementary particle. It behaves as if a point. It has momentum, and moves at light speed (implying that it cannot stop). It has an associated electromagnetic wave. The energy carried by this wave is quantized. The photon can interact with other molecules, for instance by being absorbed and emitted; with enough energy you can create them "from scratch", and they appear to always carry packets of energy with them.
One can wonder, if all particles are merely some form or arrangement of discrete bits of energy, which when aggregated in a certain way, leads to an appearance or seeming of point masses (or should I say "masses"?) and particles, and if individual Planck units of energy are really the basis for everything else in the universe, and perhaps the photon is very close to what these "energies" look like "on their own". Perhaps this is closer to what you were asking, but at this point I'm firmly into navelgazing territory, so I'll stop here.