Higher order gradients defined on a Riemmanian manifold?

Question

Let $(M,g)$ be a Riemannian manifold. I know that the gradient of a smooth function $f: M \rightarrow \mathbb R$ can be defined implicitly using the metric: $$ g(\nabla f, X) = df(X), \quad \forall X \in \Gamma(TM) $$ I have seen the higher order gradient $\nabla^k f$, $k \geq 1$ in the same context, without an explicit definition. How is $\nabla^k$ defined? By (semi-)logical concatenation of symbols, I assumed it would be a recursive definition like: $$ g(\nabla^k f, X) = (\nabla^{k-1} f)(df(X)), \forall X \in \Gamma(TM) $$ Is this right, or is the definition something else?