special relativity - Why is there a universal speed limit?

I am looking for an answer that does not rely on Special or General Relativity -and without recourse to the fact that the speed of light is frame invariant.

Is there another way of showing this universal speed limit to be necessary -one that it would have been possible to find before Einstein made his theories?

For myself I think there should be a universal speed limit because there is a law of diminishing returns when we look for ways to accelerate an object (we have to mine further and further regions of the universe which means that even an infinite universe would only have a finite amount of accessible energy)

However, following my reasoning it does not follow that this limit would be the same as the speed of light in vacuum. (I don't deny that it is identical -just that my "method" does not show this)

Answer

Backing up what zeldredge said, what you asked about is known as "relativity without light". According to the intro of this paper (arXiv link) for instance, the original argument was given as early as 1910 by Ignatowski, and has been rediscovered several times. There is a modern version due to David Mermin, in "Relativity without light", Am. J. Phys. 52, 119-124 (1984), but a pretty accessible presentation may also be found in Sec.2 of this paper by Shan Gao: "Relativity without light: A further suggestion" (academia.edu link). The basic idea is that the existence of an invariant speed follows directly from the homogeneity and isotropy of space and time, and the principle of relativity. No reference to a speed limit is needed, but it does follow that the invariant speed acts as a speed limit. The only alternative is a universe without a speed limit (infinite invariant speed), where kinematics is governed by the Galilei transformations. Why it is that our universe has a finite invariant speed, and not an infinite one, remains an open question. Gao's "further suggestion" is that the invariant speed is related to the discreteness of space and time at the Plank scale, which is an intriguing thought in its simplicity, but then it remains just a "thought" so far.