  We study fundamental rogue-wave solutions of the focusing nonlinear
Schr\"odinger equation in the limit that the order of the rogue wave is large
and the independent variables $(x,t)$ are proportional to the order (the
far-field limit). We first formulate a Riemann-Hilbert representation of these
solutions that allows the order to vary continuously rather than by integer
increments. The intermediate solutions in this continuous family include also
soliton solutions for zero boundary conditions spectrally encoded by a single
complex-conjugate pair of poles of arbitrary order, as well as other solutions
having nonzero boundary conditions matching those of the rogue waves albeit
with far slower decay as $x\to\pm\infty$. The large-order far-field asymptotic
behavior of the solution depends on which of three disjoint regions
$\mathcal{C}$, $\mathcal{S}$, and $\mathcal{E}$ contains the rescaled
variables. On the regions $\mathcal{C}$ and $\mathcal{S}$ we show that the
asymptotic behavior is the same for all continuous orders, while in the region
$\mathcal{E}$ the discrete sequence of rogue-wave orders produces distinctive
asymptotic behavior that is different from other cases.
