Could the speed of causality be (significantly) faster than $c$? The other day my son (13) asked me whether it was possible that light went very slightly slower than our best measured $c$, and at the same time had a very tiny mass, but we aren't able to measure these because they are so small. Although I told him that I didn't think that that was possible or made sense, it got me to thinking along a related trajectory:
We think of $c$ as the speed of light but really it's the speed of universal causality/information (I'll just say "causality", but you can read both). In fact, $c$ isn't just for light; it's the speed of any massless wave, right?  So what if the actual limit of causality isn't $c$, but that's just the fastest speed that we know of things happening causally in the known universe. Maybe the actual speed limit of causality is significantly faster than $c$, and we just don't know of anything that goes faster than $c$ (maybe disentanglement?)
 A: The $c$ that appears in the equations of relativity (including the famous $E = m c^2$ is the speed of causality. This is the special, unique speed that is the same for all observers regardless of their relative motions. Because it is the same for all observers, it acts as a scale factor linking space and time.
Only one such unique speed can exist, and there have been many measurements of it. It is certainly possible that this is not the speed of light; as your son suggested, it's conceivable that light has a tiny mass and hence moves slightly slower than $c$. But the relativistic $c$ (speed of causality/maximum speed) can't be significantly different from the speed of light or we would have noticed it, e.g. in particle accelerators (where things are moving very close to the maximum possible speed). And as other commentators have pointed out, there are very good theoretical reasons to suppose that light is massless and hence the speed of light is the speed of causality.
A: There is no such thing as an unambiguous "speed of causality", because causality itself is a very vague notion when you actually try to nail it down.
What is true is that the speed that we call the speed of light - but is, as you say, actually meant to be the speed of all massless particles - limits information transfer in the following sense:
For any event $x$ (i.e a time and a place) in spacetime, there is a set of events (the past lightcone and the events inside it) which can "causally" influence what happens at the event. The light cone is precisely the set of events from which something travelling at the speed of light can reach $x$ - from any event outside of it you would need to travel faster-than-light.
Since nothing - massless or massive - can travel faster than the speed of light, you might be tempted to say that therefore, of course, the speed of light is "the speed of causality"; how could anything influence something else without travelling? But the problem is, again, that our intuitive notion of what it means to be "cause and effect" - or to have "causality" - doesn't really map neatly to the physical ideas of something travelling, nor does the light-cone picture of causality alone really produce a world of "cause and effect" that we would like.
On the one hand, without travelling faster-than-light, it is possible to imagine spacetimes with so-called closed timelike curves, and something travelling along such a curve is in its own past lightcone. Cause without effect, or rather an effect that is its own cause. Is this "causality"? What is the "speed of causality" in this case? (The common answer is that such spacetimes are bad because they "violate causality".)
On the other hand, quantum theory makes everything even more complicated (as usual) - Bell's theorem tells us that either there are "effects" that propagate superluminally, or the world is not realist (for more discussion of this, see this answer of mine and this answer of mine). Crucially, which of these two to choose is a realm of metaphysics called quantum interpretations, but the predictions of quantum mechanics do not (or only in extremely contrived cases, depending on who you listen to) depend on the interpretation chosen. And so, the "speed of causality" - indeed, perhaps causality itself - is exposed as the incoherent idea that it is: In some interpretations of quantum mechanics, this speed is infinite - measurements on "one part" of a wavefunction instantaneously affect every part of this wavefunction, throughout the whole universe - and in others it is still finite, and a measurement doesn't actually have to propagate any changes at all, and there's probably all sorts of hybrid interpretations, and yet it doesn't make one bit of difference to their predictions of the world we observe.
For further ruminations on the incoherence of the idea of causation, I recommend Norton's (in)famous paper "Causation as Folk Science".
A: Firstly note that the speed of light is now defined as an exact value.  Light will travel at this speed in a vacuum simply because we defined it that way.  We used to define a meter and measure the speed, but years ago "they" (the people who define standards of measurement) decided it was more accurate to define the speed and the meter is derived from that.
So there's no way for light to travel at any other speed now.
Now there have been particles we once thought were massless, and massless particles in relativity always travel at the speed of light (photons are justy one example of massless particles).  Those particles were neutrinos, but eventually small differences in their velocity from that of light led us to understand that they had mass and travel slower than light (but still very close to light speed because their mass is so small that it takes very little kinetic energy to move them very fast).
We also have been able to compare the speed of gravity (the speed changes in gravitational fields propogate at) to the speed of light thanks to the wonderful LIGO experiments.  There's an experimental error, but at this point we have no reason to think that gravity does not propogate at the speed of light as well.  It pretty much works that way with everything we can test - if we expect it to travel at the speed of light it will within the error range we can manage with our experiment.

So what if the actual limit of causality isn't c, but that's just the fastest speed that we know of things happening causally in the known universe

All our theories would be broken, but broken in a way that would be very hard to fix because they'd still be almost right.  This would be a kind of doomsday scenario for most physicists.

Maybe the actual speed limit of causality is significantly faster than c, and we just don't know of anything that goes faster than c

Well if we don't know of it, it's not smething physics can talk about at all.  It's just wild speculation, not physics.
It's hard to imagine a way to disconnect the speed of light from the speed of causality without pretty major reworking of physics.  Again this would be rather like being told that you've to fix some major problem in your jet aircraft while at the same time not changing anything it does now apart from the new problem.  I am sure people are working on theories with variable speeds of light and all mod cons, but getting those to agree with everything we already know is the problem.  You can design any theory you like (they're published all the time), but making them work as well as existing theories is the problem.  Remember we like existing theories because they can be used to predict how stuff behaves very accurately.
It is only fair to point out that there are many alternative theories to general relativity which are either discarded, still debated or still being developed  which may provide a theoretical basis for things liek a variable speed of light.  These are very advanced research topics which I do not claim to fully understand myself.  Mainstream physics basically sticks with what they know works well, which is general relativity while (always) looking for something better.  But testing a better theory often has to wait for the technical aility to make measurements to catch up.  So we await developments.
