[geordief (OP)]

...in any system (eg the knowable universe ) whether finite or infinite?

If so what factors might  determine its value?

[Kryptid]

The speed of light? Or are you including things like the metric expansion of space where galactic recession velocities can exceed light?

[CliffordK]

A particle accelerator should be able to demonstrate that the more energy one puts into small particles, the closer they get to the speed of light without going over. 

[geordief (OP)]

The speed of light? Or are you including things like the metric expansion of space where galactic recession velocities can exceed light?

If one  goes back (or even forward) in time  to a period when all the matter in the universe is gathered in two or three (or at least a finite number of )  bodies,is it possible to show   a priori   that no matter what processes occur that alter the relative motion between these bodies that there will be a maximum velocity attainable?

I understand that particle accelerators show experimentally  that particles cannot be accelerated to c but my question is whether this experimental finding  (not the finding of the precise value   ,just the fact that a ceiling exists)  can be shown to be a logically inescapable outcome based on the ways we know that bodies can be accelerated.

[Halc]

The speed of light? Or are you including things like the metric expansion of space where galactic recession velocities can exceed light?

If one  goes back (or even forward) in time  to a period when all the matter in the universe is gathered in two or three (or at least a finite number of )  bodies

The early universe did not have matter gathered into bodies at all, and certainly not a finite number of anything.

no matter what processes occur that alter the relative motion between these bodies that there will be a maximum velocity attainable?

Motion is a complete abstract relation, so there isn't an a-priori limit on how one chooses to assign position and time to various events along a worldline. For instance, there is no limit to proper velocity, which is proper acceleration multiplied by the local time. If I accelerate at 12g for a month as measured on my watch, I will have a proper velocity of over c, which means I can get to stars 100 LY away in less than 100 years (again, as measured on my watch).
Relative to a valid inertial coordinate system, a body cannot move as fast as c. Relative to a non-inertial coordinate system (like the cosmological frame mentioned by Krypid) or say a rotating frame, there is again no limit. Betelgeuse moves at over 4000c relative to the rotating frame in which my house is completely stationary.

my question is whether this experimental finding  (not the finding of the precise value   ,just the fact that a ceiling exists)  can be shown to be a logically inescapable outcome based on the ways we know that bodies can be accelerated.

Essentially from the constancy of light speed, one can derive the velocity addition formula
u = (v+u') / (vu'/c² + 1)
Here's a site that does the derivation: https://www.sjsu.edu/faculty/watkins/relavelocities.htm
That formula shows that relative to a give inertial frame, no amount of acceleration can be applied to an already moving object that will give it a speed faster than light.
Relative to non-inertial coordinate system, there's no constant light speed, so all bets are off. Even light can be made to go fast or come to a complete stop and go back the way it came.

[Eternal Student]

Hi geordief.  I hope you are well.

   That's an extremely complicated question.

If one  goes back (or even forward) in time  to a period when all the matter in the universe is gathered in two or three (or at least a finite number of )  bodies

   I don't know that there was (or will be) a period of time when all matter was gathered into a finite number of bodies.
Some of the most current cosmological models suggest that the universe was radiation dominated at early times.  At very early times it was all radiation and no matter.   Then matter started to form but we would need a universe of only finite extent to think that there was only a finite set of chunks of matter at some time.  In an infinite universe, I'm not sure we can identify a time where there was only a finite set of matter particles.

no matter what processes occur that alter the relative motion between these bodies that there will be a maximum velocity attainable?

   This is also complicated.  For example under General Relativity,  velocity is a vector in the tangent space at a given point in the manifold of spacetime, knowing where that point was in spacetime is as important for the velocity vector as knowing it's magnitude.  This is a lot of words and gibberish, sorry.  I will paraphrase this:   

A velocity that something can have
"over here" (on the left side of the screen).

is different to a velocity that something can have

"over here" (on the right side of the screen).

If we want to compare the velocities (in some meaningful way) we need some method to move the vectors to the same place.  This is often done by a process called "parallel transport", the details aren't too important but in flat Minkowski space, it's easy.  In curved spacetime, the velocity vector is significantly altered by parallel transport.  If we are a bit carefree about defining recession velocities, then the expansion of space can (and does) cause apparent speeds greater than the speed of light.
   So I suppose I could answer your question this way:  The expansion of space can be sufficient to impart arbitrarily large speeds of recession. 

(Halc's answer appeared while I was writing this one.  I'll stop and review what's been said).

[geordief (OP)]

Relative to non-inertial coordinate system, there's no constant light speed, so all bets are off. Even light can be made to go fast or come to a complete stop and go back the way

Interesting. So even the "maximum speed of transfer of information" runs foul of non inertial observers-or is it just  that it can be lower than c but never higher than c?(bit confused as to your " Even light can be made to go fast")

[Halc]

So even the "maximum speed of transfer of information" runs foul of non inertial observers-or is it just  that it can be lower than c but never higher than c?(bit confused as to your " Even light can be made to go fast")

It has nothing to do with observers. It has to do with the abstract coordinate system of choice, and any observer, stationary, inertial, or neither has a host of coordinate systems from which to choose .

Relative to non-inertial coordinate systems, things can move faster than c, even light. Think of Betelgeuse moving at 4000c in the rotating frame of my house. If light from it didn't move that fast, observers on one side of Betelgeuse would not be able to see it, which is absurd.  If Betelgeuse is moving at exactly 4000c in that frame, then light emitted from it in the west direction would move at 4001c, at least at first, and light emitted to the east would travel at -3999c at first.

Light currently (relative to the cosmological frame) emitted from GN-z11 (a galaxy about 32 BLY away with redshift > 11) directly towards us increases its proper separation from us at about 1.3c and will never reach us. But velocity isn't relative in that frame, so that light moves at exactly c no matter which way it is pointed.
Relative to the current inertial coordinate system of the solar system, light currently emitted from GN-z11 directly towards us travels towards us at about 1c and will reach us in (SWAG) 13 billion years . I didn't work the numbers since 'our inertial frame' isn't a clearly defined thing (and for that matter, neither is its location relative to the cosmological frame).

Point is, these are all very different numbers, but all for the same observer (Earth). It isn't about the observer, it's about the coordinate system chosen.