Consider pairs $(f, g)$ of functions from the set of nonnegative integers to itself such that:
- $f(0) \geq f(1) \geq f(2) \geq \dots \geq f(300) \geq 0$,
- $f(0) + f(1) + f(2) + \dots + f(300) \leq 300$,
- for any 20 nonnegative integers $n_1, n_2, \dots, n_{20}$, not necessarily distinct, we have $g(n_1 + n_2 + \dots + n_{20}) \leq f(n_1) + f(n_2) + \dots + f(n_{20})$.

We aim to determine the maximum possible value of $g(0) + g(1) + \dots + g(6000)$ over all such pairs of functions.

The answer is $\boxed{115440}$. The construction achieving this maximum is given by:
$$f(x) = \max(24 - x, 0)$$
and
$$g(x) = \max(480 - x, 0).$$

This construction satisfies all the given conditions and achieves the maximum sum for $g(0) + g(1) + \dots + g(6000)$.

The answer is $\boxed{115440}$.