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Choice C is correct. The value of $f\left(2\right)$ is the value of $f\left(x\right)$ when $x=2$. Substituting $2$ for $x$ in the given function yields $f\left(2\right)={\left(2\right)}^3+15$, or $f\left(2\right)=8+15$, which is equivalent to $f\left(2\right)=23$. Therefore, the value of $f\left(2\right)$ is $23$.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect. This is the value of $f\left(2\right)$ when $f\left(x\right)=x\left(3\right)+15$, rather than $f\left(x\right)=x^3+15$.

Choice D is incorrect and may result from conceptual or calculation errors.