  Observations of magnetic A, B and O stars show that the poloidal magnetic
flux per unit mass has an upper bound of 10^-6.5 G cm^2/g. A similar upper
bound is found for magnetic white dwarfs even though the highest magnetic field
strengths at their surfaces are much larger. For magnetic A and B stars there
also appears to be a well defined lower bound below which the incidence of
magnetism declines rapidly. According to recent hypotheses, both groups of
stars may result from merging stars and owe their strong magnetism to fields
generated by a dynamo mechanism as they merge. We postulate a simple dynamo
that generates magnetic field from differential rotation. The growth of
magnetic fields is limited by the requirement that the poloidal field
stabilizes the toroidal and vice versa. While magnetic torques dissipate the
differential rotation, toroidal field is generated from poloidal by an Omega
dynamo. We further suppose that mechanisms that lead to the decay of toroidal
field lead to the generation of poloidal. Both poloidal and toroidal fields
reach a stable configuration which is independent of the size of small initial
seed fields but proportional to the initial differential rotation. We pose the
hypothesis that strongly magnetic stars form from the merging of two stellar
objects. The highest fields are generated when the merge introduces
differential rotation that amounts to critical break up velocity within the
condensed object. Calibration of a simplistic dynamo model with the observed
maximum flux per unit mass for main-sequence stars and white dwarfs indicates
that about 1.5x10^-4 of the decaying toroidal flux must appear as poloidal. The
highest fields in single white dwarfs are generated when two degenerate cores
merge inside a common envelope or when two white dwarfs merge by
gravitational-radiation angular momentum loss.
