  In this paper for a given Banach, possibly infinite dimensional, manifold $M$
we focus on the geometry of its iterated tangent bundle $T^rM$, $r\in
{\N}\cup\{\infty\}$. First we endow $T^rM$ with a canonical atlas using that of
$M$. Then the concepts of vertical and complete lifts for functions and vector
fields on $T^rM$ are defined which they will play a pivotal role in our next
studies i.e. complete lift of (semi)sprays. Afterward we supply $T^\infty M$
with a generalized Fr\'{e}chet manifold structure and we will show that any
vector field or (semi)spray on $M$, can be lifted to a vector field or
(semi)spray on $T^\infty M$. Then, despite of the natural difficulties with
non-Banach modeled manifolds, we will discuss about the ordinary differential
equations on $T^\infty M$ including integral curves, flows and geodesics.
Finally, as an example, we apply our results to the infinite dimensional case
of manifold of closed curves.
