What is the speed of time?

I'm going to dare to give a very brief answer that's likely not what most folks would expect, but is deeply rooted in experiment:

The speed of time is just the speed of a clock -- that is, of how fast some kind of a repeated cycle can be done.

Clocks thus only have meaning relative to each other. You can set one as a standard, then measure other by it, but you can never really define "the" time standard.

That is actually a very Einstein way of defining time -- which is to say, it's a very Mach way of defining time, since Einstein got much of his insistence on hyper-realism in defining physics quantities from Mach.

Now, most likely you thought I was going to answer that there is some kind of velocity of an object along a time axis $t$ that has "length" in much the same fashion as X or Y or Z, not in terms of cycles. That is certainly what comes to mind for me, in fact!

While viewing $t$ as having ordinary XYZ style length turns out to be an incredibly useful abstraction, it's difficult experimentally to make $t$ to behave fully like a length. The main reason is that the clock with its cycles keeps sticking in its nose and requiring that at some point, you sort of "borrow" a space-like axis from XYZ space and use that to write out a sequence of clock cycles (called proper time or $\tau$) on paper. As a result, it's not really $t$ you are drawing in those diagrams. You are instead borrowing a bit of ordinary space and mapping clock cycles onto it, making them seem like a length more through the way you represent order them than in how they actually work.

Fortunately, there is a different and more satisfying approach to the question of whether time has length, one that is suggested by special relativity, or SR. SR says in effect that XYZ space and $t$ are interchangeable, and in a very specific way. So, even though there's always a need to write out some cycles in diagrams -- proper time happens! -- you can argue that there is nonetheless a limit at which objects traveling closer and closer to the speed of light look more and more as if their time axis has been changed into a static length along some regular XYZ direction of travel.

So, by this take-it-to-the-limit kind of thinking, you can construct a more explicit concept of $t$ as an axis with XYZ-style length.

It also provides a pretty good answer to you question. Since proper time comes to an almost complete stop as an object nears the speed of light, you can say that you have in effect "stolen" the velocity of that object or spaceship through time (from your perspective or frame, not hers!) and converted it fully into a velocity through space (from your perspective).

So there is your answer: That "stolen" velocity along $t$ appears to correspond most closely with the velocity of light $c$ in ordinary space, since that is the real-space velocity at which proper time $\tau$ comes (at the limit) to a complete halt. This idea that objects "move" at the speed of light along the $t$ axis is in fact a very common assumption in relativity diagrams. It shows up for example whenever you see a light-cone diagram whose cone angle is $45^\circ$. Why $45^\circ$? Because that's the angle you get if you assume that the "velocity" of light along the $t$ axis is identical to its velocity $c$ in ordinary XYZ space.

Now, is there some slop in how that could be interpreted? You bet there is! The idea of a "velocity" in time is for example problematic in a number of ways -- just try to write it out as a derivative and you'll see what I mean. But taking such a perspective at least in terms of how to think of the issue gives a really nice simplicity to the units involved, as well as that conceptual simplicity in how to think of it. More importantly, where such simplicity keeps popping up in the representations of something in physics, it's almost certainly reflecting some kind of deeper reality that really is there.

In my opinion the term "speed of time" has no meaning.

Speed is strictly defined as dr/dt where r=sqrt(x^2+y^2+z^2). That is what we mean by "speed", the change in space during a small interval of time.

The only meaningul analogue would be dt/dr, it would not be called speed, and when relativity is taken into account it will have interesting manifestations. Without relativity is our world time which gives the time zones and one could extract from a dt/dr, complicated because of our clock definition ( as discussed in the other answers).

The word "time" means different things. In Greek there are two words for it, chronos and kairos, the former being used for seasons, hours, sequences and "the duration" of time, and the latter used for "the supreme moment" as in the moment when two people fall in love or a child is born or Biblical, when God intervenes. So because the common misunderstanding is that "time is" relative, first we would need to agree on a "time" to measure to use as a standard for the speed of time. For example, we could use the revolution of the Earth around the Sun, and it takes one year for the Earth to make a "complete rotation". But... one year, of what? Well days of course. 365 of them. And a quarter. Ok, but, 365 days... of what? Hours, of course! And before you ask, a whole shit ton of them. Ok, so which one of those is "time"? NONE OF THEM You see, from our perspective, since both the speed at which the Earth rotates around it's axis, causing days, and the speed at which the Earth revolves around the Sun, causing years, are relatively constant speeds, we as mankind have calculated the crap out of these two durations and fabricated an idea that we call "time", which is supposedly guiding and/or determining all of existence, but there is no, quantitative or observable "thing" that anyone has been able to identify and measure as "time". The closest thing to "time" is "decay" or "entropy" or "disorder", which is again relative to the thing experiencing it but not beyond our ability to calculate and therefore incorporate into our idea of a greater notion of "time". You might as well be asking, how much is $1 worth?