Given a map $f : \Bbb R^3 \to \Bbb R^3$, the induced map $\mathbf{D} f : T\Bbb R^3 \to T\Bbb R^3$ is $\mathbf{D} f (x, u) = (f(x), Df_x(u))$. The induced map $\mathbf{D}^2 f : T^2 \Bbb R^3 \to T^2 \Bbb R^3$ is $\mathbf{D}^2 f(x, u_1, u_2, v) = (f(x), Df_x(u_1), Df_x(u_2), Df_x(v, u_2))$. Let me try the third iteration:
Sorry, I meant $D^2 f_x(v, u_2)$ in the last coordinate in the last line.
This is the Hessian, $v^T Hf^i_x u_2$, on each coordinate $f^1, f^2, f^3$ of $f$
So the third iteration is $\mathbf{D}^3 f : T^3 \Bbb R^3 \to T^3 \Bbb R^3$, let's denote an element of $T^3 \Bbb R^3$ lying over the point $x \in \Bbb R^3$ as $(x, u_1, u_2, v_1, u_3, v_2, v_3, w)$ where $u$'s are tangent vectors, $v$'s are double tangent vectors aka tangent vectors on $T_x \Bbb R^3$ i.e., in $T_u T_x\Bbb R^3$ aka variations, and $w$'s are triples tangent vectors, aka tangent vectors on $T_u T_x \Bbb R^3$, i.e., in $T_v T_u T_x \Bbb R^3$
$v$'s are variations of $u$'s and $w$'s are variations of $v$'s which are variations of $u$'s. Note the difference between on and in I have emphasized for clarity
Explicitly, this seems to be $\mathbf{D}^3 f : T^3 \Bbb R^3 \to T^3 \Bbb R^3$, given by $$\mathbf{D}^3 f(x, u_1, u_2, v_1, u_3, v_2, v_3, w) \\ = (f(x), Df_x(u_1), Df_x(u_2), D^2 f_x(v_1, u_2), Df_x(u_3), D^2 f_x(v_2, u_3), D^2 f_x(v_3, u_3), D^3 f_x(w, v_3, u_3))$$
