  Outer automorphism groups of RAAGs, denoted $Out(A_\Gamma)$, interpolate
between $Out(F_n)$ and $GL_n(\mathbb{Z})$. We consider several vastness
properties for which $Out(F_n)$ behaves very differently from
$GL_n(\mathbb{Z})$: virtually mapping onto all finite groups, SQ-universality,
virtually having an infinite dimensional space of homogeneous quasimorphisms,
and not being boundedly generated.
  We give a neccessary and sufficient condition in terms of the defining graph
$\Gamma$ for each of these properties to hold. Notably, the condition for all
four properties is the same, meaning $Out(A_\Gamma)$ will either satisfy all
four, or none. In proving this result, we describe conditions on $\Gamma$ that
imply $Out(A_\Gamma)$ is large.
  Techniques used in this work are then applied to the case of McCool groups,
defined as subgroups of $Out(F_n)$ that preserve a given family of conjugacy
classes. In particular we show that any McCool group that is not virtually
abelian virtually maps onto all finite groups, is SQ-universal, is not
boundedly generated, and has a finite index subgroup whose space of homogeneous
quasimorphisms is infinite dimensional.
