Since $0 < m < n$, we have $m + n > n > m > 0$.

Testing each expression:

• $\frac{m+n}{m} = 1 + \frac{n}{m} > 1 + 1 = 2$ (since $n > m$)
• $\frac{m+n}{n} = \frac{m}{n} + 1 < 2$ (since $m < n$)
• $\frac{m+n}{m+n} = 1$
• $\frac{m}{m+n} < 1$
• $\frac{n}{m+n} < 1$

$\frac{m+n}{m}$ has the greatest value.