Lets take a vector field $X$ on some Riemannian manifold $(\mathcal{M},g)$ with Levi-Civita connection $\nabla$. Then, the components of its covariant derivative in coordinates are
$$\nabla_{\alpha}X^{\beta}=\partial_{\alpha}X^{\beta}+\Gamma_{\alpha\gamma}^{\beta}X^{\gamma}$$
Now, since $\nabla X$ is again tensor field, we can apply a second covariant derivative, which in coordinates yields
$$\nabla_{\gamma}\nabla_{\alpha}X^{\beta}:=\nabla_{\gamma}(\nabla_{\alpha}X^{\beta})=\partial_{\gamma}(\nabla_{\alpha}X^{\beta})-\Gamma_{\gamma\alpha}^{\delta}\nabla_{\delta}X^{\beta}+\Gamma_{\gamma\delta}^{\beta}\nabla_{\alpha}X^{\delta}=…$$
Now, instead of writing it like this, let us formally change the order in which the covariant detivative act, i.e. let us write
$$\nabla_{\gamma}\nabla_{\alpha}X^{\beta}=\nabla_{\gamma}(\nabla_{\alpha}X^{\beta})=\nabla_{\gamma}(\partial_{\alpha}X^{\beta}+\Gamma_{\alpha\gamma}^{\beta}X^{\gamma})=\nabla_{\gamma}(\partial_{\alpha}X^{\beta})+\nabla_{\gamma}(\Gamma_{\alpha\delta}^{\beta}X^{\delta})=…$$
Now, mathematically speaking, the two terms on the right-hand side are ill-defined and do not make sense, since $\nabla$ is an operation acting on tensor fields and neither $\partial_{\alpha}X^{\beta}$ nor $\Gamma_{\alpha\delta}^{\beta}X^{\delta}$ are the components of a tensor field. However, if we treat these two terms as if they were rank (1,1) tensor fields, i.e. elements of $\Gamma^{\infty}(T\mathcal{M}\otimes T^{\ast}\mathcal{M})$, and use the standard formula for the connections, we will find the same and correct result for the components of $\nabla^{2}X$.
Now, this seem to work in general, i.e. for arbitrary rank tensors and an arbitrary amount of covariant derivatives (at least I never have seen a counter example).
My question, or lets say, my curiosity, is:
Why does the second "approach", in which we produce ill-defined terms in the steps in-between, work?
Is there any mathematical reason? For example, maybe one can extend the covariant derivative to more general "objects with indices" in a unique way, such that the steps in between become well-defined. For example, one can extend the covariant derivative to a map acting on tensor densities (i.e. sections of tensor product of a tensor bundle and a density bundle). Maybe there is a similar and more general notion which also includes objects like the partial derivative and the Christoffel symbols.
Best Answer
The notion that Christoffel symbols are not tensor fields is not entirely true, or at least somewhat deceptive. If we choose a set of local coordinates $x^\alpha$, then the corresponding Chrisotoffel symbols $\Gamma^\alpha{}_{\beta\gamma}$ represent a perfectly well-defined local tensor field. Issues only arise when we try to interpret the objects involved as global, coordinate-independent objects, since a different set of coordinates will not generally result in the same tensor field, even on their common domain.
In this particular computation, we are fixing a coordinate system at the start, and finding an expression for $\nabla\nabla X$ in terms of the Christoffel symbols of those coordinates, so we can interpret expressions like $\nabla_\alpha(\Gamma^{\beta}{}_{\gamma\delta})$ as a covariant derivative of a tensor field without issue.