  This article is concerned with the representation growth of profinite groups
over finite fields. We investigate the structure of groups with uniformly
bounded exponential representation growth (UBERG). Using crown-based powers we
obtain some necessary and some sufficient conditions for groups to have UBERG.
As an application we prove that the class of UBERG groups is closed under split
extensions but fails to be closed under extensions in general. On the other
hand, we show that the closely related probabilistic finiteness property
$PFP_1$ is closed under extensions. In addition, we prove that profinite groups
of type $FP_1$ with UBERG are always finitely generated and we characterise
UBERG in the class of pro-nilpotent groups.
  Using infinite products of finite groups, we construct several examples of
profinite groups with unexpected properties: (1) an UBERG group which cannot be
finitely generated, (2) a group of type $PFP_\infty$ which is not UBERG and not
finitely generated and (3) a group of type $PFP_\infty$ with superexponential
subgroup growth.
