Abstract. In this paper for a given Banach, possibly infinite dimensional, manifold M we focus on the geometry of its iterated tangent bundle T r M , r ∈ N ∪ {∞}. First we endow T r M with a canonical atlas using that of M . Then the concepts of vertical and complete lifts for functions and vector fields on T r M are defined which they will play a pivotal role in our next studies i.e. complete lift of (semi)sprays. Afterward we supply T ∞ M with a generalized Fréchet 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 ∞ M . Then, despite of the natural difficulties with non-Banach modeled manifolds, we will discuss about the ordinary differential equations on T ∞ 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.