In this lecture by Frederic Schuller, it is discussed how we can turn the tangent bundle could be turned into a manifold through charts of the manifold. So, this made me wonder, couldn't we repeat this process again on the tangent bundle infinitely and keep layering levels of tangent bundle? Is there any practical use of this viewpoint?
One thing to note is that it's pretty clear that the dimension of the tangent bundle rises by $2^j r$ where $r$ is the dimension of initial manifold, and j is the number of times we took the tangent bundle. So, for instance tangent bundle of manifold is $2r$ , tangent bundle of the tangent bundle of manifold is of dimension $2^2r$ and so on
You can absolutely do this. The tangent bundle of the tangent bundle $TTM$ is known as the double tangent bundle and appears ins some contexts. The higher order tangent bundles $T^3M$ etc. are also perfectly well-defined.
That said, applying the tangent functor multiple times is not something that happens commonly. The higher tangent bundles encode higher derivatives, but in a very complicated way. Other objects such as connections or jets are generally preferred as a way of studying higher order derivatives from a global perspective.