Using logarithmic rules, we see that \[\log_{a}a-\log_{a}b+\log_{b}b-\log_{b}a = 2-(\log_{a}b+\log_{b}a)\]\[=2-\Big(\log_{a}b+\frac {1}{\log_{a}b}\Big)\] Since $a$ and $b$ are both positive, using AM-GM gives that the term in parentheses must be at least $2$, so the largest possible values is $2-2=\boxed{0}.$