We have: 

\(2007^{2008}=2007^{4.502}=\left(\overline{...1}\right)^{502}=\overline{...1}\)

\(2008^{2007}=2008^{4.501+3}=\left(\overline{...6}\right)^{501}+2008^3=\overline{...6}+\overline{...2}=\overline{...8}\)

\(\Rightarrow2007^{2008}+2008^{2007}=\overline{...1}+\overline{...8}=\overline{...9}\)

So the ones digit is 9