Putnam 2025 B6

Let $\mathbb{N} = \{1,2,3,\dots\}$. Find the largest real constant $r$ such that there exists a function $g\colon \mathbb{N} \to \mathbb{N}$ such that $$g(n+1)-g(n) \geq (g(g(n)))^r$$ for all $n \in \mathbb{N}$.