Central limit theorem - num random variables vs. sample size?

Your question shows some confusion! A sample is usually represented by a sequence of random variables iid $X_1, X_2, dotsc, X_n, dotsc$, where each random variable is a function from some sample space $omega mapsto X(omega)$, the argument $omega$ often omitted from the notation. So a realization is only one value, always. So $n$ in the CLT refers to the number of observations, each observation represented with one random variable.

So in your chisquare example, $n$ has nothing to do with the number of degrees of freedom $nu$. For the iid CLT, the only assumption is that expectation and variance exists as finite numbers. Since that is the case for the chisquare distribution, the CLT applies and when the number of observations $n toinfty$, the standardized mean converges to a normal distribution.