According to the Bohr Theory, which transition in the hydrogen atom will give rise to the least energetic photon?

The energy of a photon $\Delta E\propto \left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)$. As $n$ increases, the energy gap between consecutive levels decreases. Thus, $n=6\to 5$ has a smaller energy difference than $n=5\to 4$, and significantly less than transitions to lower shells like $n=1$ or $3$.