  We study linear and algebraic structures in sets of bounded holomorphic
functions on the ball which have large cluster sets at every possible point
(i.e., every point on the sphere in several complex variables and every point
of the closed unit ball of the bidual in the infinite dimensional case). We
show that this set is strongly c-algebrable for all separable Banach spaces.
For specific spaces including lp or duals of Lorentz sequence spaces, we have
strongly c-algebrability and spaceability even for the subalgebra of uniformly
continuous holomorphic functions on the ball.
