Non-Existence for Forward-Index Multiplicative Recurrences

The problem was motivated by this related MSE question, although the recurrence here is structurally different.

Let $f, g : \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ satisfy

$$f(n) \ge n+1, \qquad g(n) \ge (1+\varepsilon)n$$ for some fixed $\varepsilon > 0$. Consider

$$a_{n+2} = a_{n+1} \bigl(1 + a_{f(n)}a_{g(n)}\bigr), \qquad n \ge 1,$$ where $a_n \in \mathbb{Z}_{>0}$.

I can prove that no infinite sequence of positive integers satisfies this recurrence.

Indeed,

$$a_{n+2} \ge a_{n+1}^2,$$ so $a_n \ge 2^{2^{n-3}}$ for $n \ge 3$. Evaluating the recurrence at $m = g(n) - 2$, for sufficiently large $n$ we have

$$g(g(n)-2) \ge g(n) \qquad \text{and} \qquad f(g(n)-2) \ge g(n)-1.$$ Hence monotonicity gives

$$a_{g(n)} \ge a_{g(n)-1}^2 a_{g(n)},$$ so $a_{g(n)-1} = 1$, contradicting the growth bound. $\blacksquare$

The natural question is now how sharp the hypothesis on $g$ is.

For example, what happens if the uniform expansion condition is weakened to

$$g(n) \ge n+1?$$ The extreme case $g(n) = n+1$ is no longer covered by the above argument. For instance, taking $f(n) = g(n) = n+1$ gives

$$a_{n+2} = a_{n+1} + a_{n+1}^3,$$ which clearly has infinitely many positive-integer solutions.

Main Question: How far can the condition $g(n) \ge (1+\varepsilon)n$ be weakened while non-existence is still guaranteed?

Sub-Question: In particular, what happens for sub-linear expansion functions such as:

1. $g(n) = n + C$

2. $g(n) = n + \lfloor\sqrt{n}\rfloor$

3. $g(n) = n + \lfloor\log n\rfloor$?