There's a lower bound of $2^n$ that comes immediately from volume considerations. For fixed $n$ I'm not sure when the upper bound is known exactly (this seems uncomfortably close to the Kepler conjecture, which was hard even in $3$ dimensions), but there's asymptotic results for covering space (or spheres of growing radii) by spheres going back to Rogers ("A Note on Coverings", 1957...see also this paper of Ilya Dumer) that might be able to get you something like $3^n n \log n$. I'm not familiar enough with this area to say if there's more specific results pertaining to your problem.
A quick (not at all optimal, and not my own, though I don't know a source to give for it) argument can get you a bound of $5^n$. Consider the following process: Place balls of radius $1/2$ centered at $x_1, x_2, \dots$ in the larger sphere, subject only to the constraint that no two of these balls intersect (these balls can lie partially outside the sphere, but they must be centered inside the sphere). Continue placing these balls in an arbitrary fashion until no more such balls can be placed.
At this point all the balls you have placed are disjoint and lie in a sphere of radius $5/2$ (the extra $1/2$ coming from balls which are centered near the edge of the original sphere). Therefore there are at most $5^n$ balls placed. Furthermore, each point in the sphere is within $1$ of one of your ball's centers, since otherwise you could fit another ball of radius $1/2$ in. So replacing your radius $1/2$ balls with radius $1$ balls with the same center gives you a covering.