smc

$k\equiv 2008^2+2^{2008}\equiv 8^2+2^4\equiv 4+6\equiv 0(\mathrm{mod}10)$. So, $k^2\equiv 0(\mathrm{mod}10)$. Since $k=2008^2+2^{2008}$ is a multiple of four and the units digit of powers of two repeat in cycles of four, $2^k\equiv 2^4\equiv 6(\mathrm{mod}10)$. Therefore, $k^2+2^k\equiv 0+6\equiv 6(\mathrm{mod}10)$. So the units digit is $6\Rightarrow \boxed{6}$.