  It is well known that Akhmediev breathers of the nonlinear cubic Schrodinger
equation can be superposed nonlinearly via the Darboux transformation to yield
breathers of higher order. Surprisingly, we find that the peak height of each
Akhmediev breather only adds {\it linearly} to form the peak height of the
final breather. Using this new peak-height formula, we show that at any given
periodicity, there exist a unique high-order breather of maximal intensity.
Moreover, these high-order breathers form a continuous hierarchy, growing in
intensity with increasing periodicity. For any such higher-order breather, a
simple initial wave function can be extracted from the Darboux transformation
to dynamically generate that breather from the nonlinear Schrodinger equation.
