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I have been curious to find what could be the significance of Planck force? It is calculated by the formula $c^4/G$, where $c$ is the speed of light; $G$ is the gravitational constant. Thus (the speed of light^4) / gravitational constant = 1.21031359 $\times 10^{44}$ newtons.

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There is no particular physical significance; it's just a unit. Of course, in any system where such a large force is exerted, out current theories should not be accurate, and a quantum theory of gravity or some as-yet-unknown theory would be needed to accurately describe its behavior.

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