Let's say we denote by $T^{(n,m)}M$ the vector-bundle of rank $(n,m)$ tensors on a manifold $M$ and by $\Lambda^pM$ the vector-bundle of $p$-forms on $M$. Is there a relationship (perhaps a diffeomorphism to some sort of direct sum expansion) between the various $\Lambda^p M$ (or the various $T^{(n,m)}M$) and iterated $TM$'s or $T^\*M$'s? Intuitively, for example, the tensors of rank $2$ are like 2nd order terms in a Taylor expansion -- but so is $TTM$. Using physics notation, where $M$ is parametrized locally by coordinates $q$, $TM$ looks like $(q,\dot{q})$ and $TTM$ look like $(q,\dot{q},\dot{q}',\ddot{q})$ where I use the ' to denote a 2nd copy of the $\dot{q}$. If $M$ is $r$ dimensional then $TM$ has dimension $2r$, $TTM$ has dimension $4r$, etc, and $\Lambda^p M$ has dimension $({r\atop p})+r$. Is there some sort of useful morphism between $\Lambda^pM$ and a direct sum of $T^\*M$, $T^\*T^\*M$, etc? Ditto for the $T^{(n,m)}M$? It feels like there should be some relationship between tensor bundles (or form-bundles) and iterated tangent or cotangent bundles, but I can't seem to find one (or determine an obvious ways to construct a series that even would have the correct dimension). On a similar note, are there any such relationships amongst the iterated $TM$'s and $T^\*M$'s themselves? Thanks in advance for any insights. 



Is there a relationship between tensor (or form) bundles and iterated tangent/cotangent bundles on a manifold?
The answer is 'basically, no'. The tensor bundles that you list, such as $T^{(n,m)}M$ and its quotients (such as $\Lambda^p(TM)$, etc.), are *first order* prolongations of $\mathrm{Diff}(M)$, whereas $TTM$ is a *second order* prolongation of $\mathrm{Diff}(M)$. All of these are examples of functors from the category $\mathsf{Diff}\_n$ whose objects are smooth $n$-manifolds and whose morphisms are diffeomorphisms into some $\mathsf{Diff}\_m$. For example, the tangent bundle functor $\mathsf{T}:\mathsf{Diff}\_n\to \mathsf{Diff}\_{2n}$ and the cotangent functor $\mathsf{T^\ast}:\mathsf{Diff}\_n\to \mathsf{Diff}\_{2n}$ are first order while $\mathsf{T\circ T}:\mathsf{Diff}\_n\to \mathsf{Diff}\_{4n}$ is second order. For this reason, this composition cannot be written as a first order functor, i.e., it has no natural transformation that makes it equivalent to a first order functor, such as a sum of tensor bundles (which would be a first order functor).


You can see this very concretely in the case $n=1$, if you just write out what is happening in local coordinates.


