Given a smooth map $f : M \to N$ between manifolds, the differential gives a map $Tf : TM \to TN$ between tangent bundles. Taking another differential gives a map $T^2f : T^2M \to T^2N$ between tangent bundles of tangent bundles.
This answer gives one way we can think about $T^2f$. Namely, in the same way that the elements of $TM$ are represented by curves $\gamma(t) : (-\varepsilon, \varepsilon) \to M$, the elements of $T^2M$ are represented by rectangles $\gamma(s, t) : (-\varepsilon, \varepsilon) \times (-\varepsilon, \varepsilon) \to M$. Then, given a chart $\varphi : U \to \mathbb{R}^m$ of $M$, we obtain a chart $T^2\varphi$ of $T^2M$ given by $$ T^2\varphi[\gamma] = \left((\varphi \circ \gamma)(0, 0), \frac{\partial(\varphi \circ \gamma)}{\partial s}(0, 0), \frac{\partial(\varphi \circ \gamma)}{\partial t}(0, 0), \frac{\partial^2(\varphi \circ \gamma)}{\partial s\partial t}(0, 0)\right). $$ Given a chart $\psi$ of $M$, we then have $$ (T^2\psi \circ T^2f)[\gamma] = \left((\psi \circ f \circ \gamma)(0, 0), \frac{\partial(\psi \circ f \circ \gamma)}{\partial s}(0, 0), \frac{\partial(\psi \circ f \circ \gamma)}{\partial t}(0, 0), \frac{\partial^2(\psi \circ f \circ \gamma)}{\partial s\partial t}(0, 0)\right). $$ By the chain rule, we obtain the concrete expression $$ (T^2\psi \circ T^2f \circ T^2\varphi^{-1})(x, v, w, \theta) = \left(g(x), \sum_{i = 1}^m \frac{\partial g}{\partial x_i} v_i, \sum_{i = 1}^m \frac{\partial g}{\partial x_i} w_i, \sum_{i, j = 1}^m \frac{\partial^2 g}{\partial x_i \partial x_j}v_iw_j + \sum_{i = 1}^m \frac{\partial g}{\partial x_i}\sigma_i\right), $$ where $g = \psi \circ f \circ \varphi^{-1}$. I've denoted the fourth coordinate with a Greek letter since it seems to represent something distinct from $x$ (which is a point) and $v, w$ (which are vectors).
These calculations shows that $T^2 f$ encodes information about the second partial derivatives of $f$. But the way it does so is distinct from, say, the Hessian, which ignores the $\sigma$ component. Of course, this must be the case since there is no coordinate-free way to define the Hessian. That said, there does seem to be some relation between the Hessian and $T^2 f$, as both have coordinate expressions involving the sum $\sum_{i, j} (\partial^2 f / \partial x_i \partial x_j)v_iw_j$ plus another term.
What exactly is the map $T^2 f$? Is it related to some more familiar notion such as the Hessian, or perhaps jets? If not, why doesn't $T^2 f$ show up more often when studying manifolds?
