  Polish doughnuts (PDs) are geometrically thick disks that rotate with
super-Keplerian velocities in their innermost parts, and whose long and narrow
funnels along rotation axes collimate the emerging radiation into beams. In
this paper we construct an extremal family of PDs that maximize both
geometrical thickness and radiative efficiency. We then derive upper limits for
these quantities and subsequently for the related ability to collimate
radiation. PDs with such extreme properties may explain the observed properties
of the ultraluminous X-ray sources without the need for the black hole masses
to exceed ~ 10 solar masses. However, we show that strong advective cooling,
which is expected to be one of the dominant cooling mechanisms in accretion
flows with super-Eddington accretion rates, tends to reduce the geometrical
thickness and luminosity of PDs substantially. We also show that the beamed
radiation emerging from the PD funnels corresponds to "isotropic" luminosities
that linearly scale with the mass accretion rate, and do not obey the familiar
and well-known logarithmic relation.
