What is the slowest possible speed? According to special relativity, nothing can go faster than the speed of light, and nothing can be distinguished to be in a state of absolute rest.  
So it makes me wonder: is there a slowest possible speed?  
Right now I'm wanting to think that there isn't.  I'm pondering the idea that no matter what speed you're going, there could very well be some object (1) that is going a relativistic speed faster than an object (2) in your local reference frame, and at the same time, you going at a relativistic speed faster than another object (3). True, something moving in the opposite direction at a fast speed may appear to be going slower than you, and I'm not sure how that fact affects my hypothesis.  I'd like to see others' analysis of this "no slowest possible speed" idea to see whether I'm on the right track or just way off.
It's interesting to think that there may be a fastest speed and no slowest speed, when our intuition gives us the opposite inclination.
 A: The slowest speed possible with respect to your reference frame is achieved by you, regardless of special relativity.

...in your local reference frame, and at the same time, you going at a relativistic speed faster than another object...

I'm not quite sure how to interpret this. By definition, in your local reference frame, your speed = 0.
I think you're misinterpreting special relativity.

nothing can be distinguished to be in a state of absolute rest.

This does not mean you cannot have a speed = 0 from a particular reference frame, but rather that unless you are moving at c, your velocity will change between reference frames.  
A: 
According to special relativity, nothing can go faster than the speed
  of light,

The speed of matter and the propagation of information is believed to be limited by the speed of light.

and nothing can be distinguished to be in a state of
  absolute rest.

Here you mix in the concept of absolute frames of reference.
There is also no absolute speed of 80 miles per hour.
Of course there is the absoulute speed of light. :-)
So for matter it depends on the frame of reference what speed you attribute to it.

So it makes me wonder, is there a slowest possible
  velocity? 

A speed of zero is perfectly legal for matter, as you can attach a frame of reference to it it, where it rests.
Light does not rest. 

Right now I'm wanting to think that there isn't. I'm
  pondering the idea that no matter what speed you're going, there could
  very well be some object (1) that is going a relativistic speed faster
  than an object (2) in your local reference frame,
  and at the same
  time, you going at a relativistic speed faster than another object
  (3). True, something moving in the opposite direction at a fast speed
  may appear to be going slower than you, and I'm not sure how that fact
  affects my hypothesis. 

That would be the case, if velocities $v_1$ and $v_2$ add up like 
$v_1 + v_2$.
However for matter that is not the case, at speeds close to the speed
of light, the addition of velocities is notably different (link).
In the end matter is bound to travel at a speed $0 \le v < c$.
And for light you can do what you want, it travels at $c$ (in vacuum).

I'd like to see others' analysis of this "no
  slowest possible speed" idea to see whether I'm on the right track or
  just way off.
It's interesting to think that there may be a fastest speed and no
  slowest speed, when our intuition gives us the opposite inclination.

Matter can be at rest ($v=0$). No problem here. 
A: Light does not have an absolute speed. It has the same speed in any frame of reference. These are not the same thing.
Regarding the other question:
Is it possible for an object to have the same speed in all frames of reference that's actually different from the speed of light?
Consider this argument (modified from this page):
Suppose an object A is moving with a velocity v relative to an object B, and A is moving with a velocity w relative to object C.  Finally, B is moving with a velocity u relative to an object C.
What speed must v have so w = v?  This will give the answer to our question.
                         v
               u      -------> A
            -------> B
           C        w
            ----------------->

In non-relativistic mechanics the velocities are simply added and the answer is that A is moving with a velocity w = u+v relative to C.  But in special relativity the velocities must be combined using the formula
$$ w = \frac{u+v}{1+\frac{uv}{c^2}}$$
We just need to set w = v, and solve for v:
$$ v = \frac{u+v}{1+\frac{uv}{c^2}}$$
$$ v + \frac{uv^2}{c^2} = u + v$$
$$ \frac{v^2}{c^2} = 1$$
$$ v^2 = c^2$$
$$ v = c$$
This is the only solution, and it is true for all u.
Well, not exactly, note that if u = 0, we divided by 0, so we should make this a special case.  But in that case
$$ w = \frac{0 + v}{1 + 0} = v$$
So, the only way where an object can have the same velocity in two inertial reference frames is if the relative velocity of the reference frames is zero or where the object is traveling at the speed of light in both reference frames.
Also, note that if an object is going the speed of light in one reference frame, that it is going the speed of light in all reference frames is easily derived from this equation.
Replace v with c and see that w must also equal c:
$$ w = \frac{u+c}{1+\frac{uc}{c^2}}$$
$$ w = \frac{u+c}{1+\frac{u}{c}} = \frac{1}{\frac{1}{c}}\cdot\frac{u+c}{c+u} = c$$
A: When pondering the question of the slowest possible speed, I initially found myself agreeing with all of the other given answers. There is no absolute rest velocity because there is no absolute rest frame. That means you will never find a reference frame with a velocity that can be boosted to and then boosted back from that would allow you to age faster (not sure why you'd want to). You definitely age the fastest at rest in your own inertial frame (ignoring one-way trips to other inertial frames).
But then I thought, "Maybe there's no slowest local velocity, but is there a slowest global velocity? One where even an alien race in a distant galaxy would be likely to determine their velocity wrt to it?" The answer to that question is yes. There is a universally agreeable rest velocity and that comes to us courtesy of the cosmic microwave background (CMB). See, the CMB is pretty much isotropic to any comoving observer. A comoving observer is one that is not moving wrt local space. The only motion a comoving observer has is its recessional velocity, which is due to the expansion of space and isn't really influenced by special relativity. This would, therefore, make an excellent definition of the absolutely lowest possible speed. The measure of an object's peculiar velocity, which is the velocity that it has different from a comoving object, would be something even distant alien races could agree on as a measure of how fast something is moving locally.
When something has a non-zero peculiar velocity, that causes the CMB that it observes to doppler shift. We can also measure the peculiar velocity of distant objects by finding the difference in the redshift of light from them from what we'd expect a comoving object at that distance to have. This means we can measure our velocity relative to local background space and the velocity of distant objects relative to the distant local background space and this number is something everyone would agree on (our peculiar velocity is around 0.0012c).
So while it's true you could always find a frame of reference where something else is moving slower than an object with 0 peculiar velocity, I daresay you won't find such a universally agreeable frame of reference to use. As long as we can assume the CMB is isotropic for a comoving observer, then this becomes a very natural frame of reference to use to say "the slowest possible velocity is one with no peculiar velocity". Even if that's just imposing a preferred frame of reference, it's an agreeable one to impose.
TL;DR There's not really an absolute rest velocity, but there is a universally agreeable rest velocity that we can use just the same.
A: Answer
I feel strongly there is no way to prove that such thing as a "slowest possible non-zero speed" exists. However there are realistic limits regarding the degree to which slower and slower speeds can reliably be measured.
Example (purely hypothetical)
Consider a pair of neutrinos, ν1 and ν2, emitted from the sun, one right after the other, at times T0 and T1, where:
T1 - T0 = tp (1 Planck time)
Now consider that they travel along the exact same path over the course of 500 billion trillion quadrillion years, and eventually, they arrive at some distant galaxy, where some aliens have a neutrino detector set up. When they get there, they arrive at times T0' and T1', where:
T1' - T0' = 2tp (2 Plank times)
What is the relative velocity difference between the two neutrinos when they were emitted from the sun? It's a really darn small number. 
This number could continue getting smaller the farther away the alien detector was, assuming T1' - T0' remained 2tp. Therefore there does not seem to be a minimum possible speed.
Discussion
Clearly this hypothetical example is ridiculous. I made it that way to illustrate a point, which is that determining a velocity requires two measurements, and it requires being able to identify something as being the same thing you observed both times. 
In our example the aliens would have no way to know if the neutrinos both came from exactly the same place or not. Even if they did they'd have no way to observe when the neutrinos were emitted relative to one another.
I have a very strong feeling that the minimum speed that it is possible to observe depends heavily upon the detection equipment, and even then, there are realistic limits to consider. 
For example, are we talking about trying to measure the relative speed difference between two objects that are moving at relativistic speeds compared to us? Or are we staring at an apparently immobile object and waiting a million years for it to move a single plank length?
I think there comes a time when an observer would either give up and say, "It ain't moving," or they would die. 
Furthermore, instruments used to detect speed are generally calibrated to units of time such as meters per second with a limited degree of accuracy. Something moving 1 plank length per hundred googolplex years would not be considered "in motion," would it? 
To put it another way the minimum possible measurable non-zero speed is lp/∂T, where ∂T is the change in time from the first point the object was measured at to the second point the object was measured at, and lp is a planck length.
