Does the transitivity of big-O notation hold for asymptotically nonnegative functions?
Suppose $f(n)$, $g(n)$ and $h(n)$ are asymptotically non-negative functions such that $f(n) = O(g(n))$ and $g(n) = O(h(n))$.

Then, by definition, there exist positive constants $c_1, c_2, m_1, m_2$ such that

$$
0 \le f(n) \le c_1 g(n)
$$

whenever $n \ge m_1$ and

$$
0 \le g(n) \le c_2 h(n)
$$

whenever $n \ge m_2$.

Choose $c = c_1 c_2$ and $n_0 = \max\{m_1, m_2\}$. 
Note that both $c$ and $n_0$ are positive.
Then, whenever $n \ge n_0$, we have

$$
0 \le f(n) \le c_1 g(n) \le c_1 c_2 h(n) = c h(n)
$$

which implies that $f(n) = O(h(n))$.