For which values of $k$ is there an $X$ with $\Omega^kX \cong X$? Bott periodicity can be formulated as $\Omega^2 U \cong U$ where $\Omega$ denotes the based loop space functor and $U$ is the direct limit of unitary groups. The real version can be formulated as $\Omega^8 O \cong O$ where $O$ is the direct limit of orthogonal groups.

Are there other spaces which are homotopy equivalent to an iterated loop space of themselves? For which values of $k$ is there an $X$ such that $\Omega^kX \cong X$ (but $\Omega^iX \not\cong X$ for $1 \leq i < k$)?


There are always such spaces, for silly reasons: you can take $X$ to be a periodic product of Eilenberg-MacLane spaces. 
All of the natural examples I know have even period, though. 
