  We unveil a mechanism enabling a fundamental rogue wave, expressed by a
rational function of fourth degree, to reach a peak amplitude as high as a
thousand times the background level in a system of coupled nonlinear
Schrodinger equations involving both incoherent and coherent coupling terms
with suitable coefficients. We obtain the exact explicit vector rational
solutions using a Darboux-dressing transformation. We show that both components
of such coupled equations can reach extremely high amplitudes. The mechanism is
confirmed in direct numerical simulations and its robustness confirmed upon
noisy perturbations. Additionally, we showcase the fact that extremely high
peak-amplitude vector fundamental rogue waves (of about 80 times the background
level) can be excited even within a chaotic background field.
