The answer to your question, as written in the details, is no for all known particles. Empirical evidence comes from the fact that in particle colliders like the Large Hadron Collider but also its many earlier antecedents, beams of protons and heavy nuclei - composed of protons and neutrons - can be accelerated arbitrarily close to, but no faster than, the speed of light. These particles are not photons, and do not contain electrons anywhere in their composition - they are instead blobs of quarks. Moreover, muons - a particle similar to the electron in some ways but equally an elementary particle that is not composed of anything, including electrons - provide less direct tests for this idea.
The reason that this is to be expected is that it is not so much that the "speed of light" arises from "interaction between electrons and photons" - which I believe you're using to intend to really mean the electromagnetic interaction - but rather it comes from the geometry of spacetime, and in fact it is this geometry of spacetime which shapes electromagnetic interactions, not the other way around. In particular, it both limits the speed of light and creates the possibility of magnetic fields. So yes, your last hunch at the end of your post is, to the best of our knowledge, correct.
The key fact about the geometry of spacetime is that it is "Lorentzian", which means that the mathematical relations describing how the point of view of an observer changes when sie changes speeds - which effectively amount to its "rotations" - are such that there is a particular speed which is not changed by them, and this speed is what we call the speed of light. This speed is not necessarily a limit, but the Lorentzian geometry also tells you that if you could somehow exceed it, you would be able to travel or at least send messages backwards in time. The facts that we have seen nothing to do so, including that we have received no messages from our future selves, that such backwards-in-time travel creates interesting logical problems like the so-called "grandfather paradox", and moreover the sheer mountain of experimental evidence which verifies to very high precision (including all tests of the universality of the light speed limit but also of related consequences of the geometry like that of time dilation, which the aforementioned muon studies deal with) the validity of the Lorentzian geometry as being the geometry on which our Universe is based, are all good reasons to consider the "speed of light" as the absolute maximum value allowed for the speed of any moving objects in the real Universe.