Let $F$ be a nonabelian free finitely generated group, and $F = G_0 \rhd G_1 \rhd G_2 \dots$ a strictly descending subnormal chain of subgroups ($G_n \lhd G_{n-1}$ for each $n \in \mathbb{N}$) each closed in $F$ (with respect to the profinite topology). Set $$H = \bigcap_{n \in \mathbb{N}} G_n$$

Is it possible for $H$ to be nontrivial and finitely generated?

Since it seems possible, I further assume that $[F : G_n] < \infty$ for $n \in \mathbb{N}$ making the assumption on being closed in the profinite topology superfluous.