  In the system of a gravitating Q-ball, there is a maximum charge $Q_{{\rm
max}}$ inevitably, while in flat spacetime there is no upper bound on $Q$ in
typical models such as the Affleck-Dine model. Theoretically the charge $Q$ is
a free parameter, and phenomenologically it could increase by charge
accumulation. We address a question of what happens to Q-balls if $Q$ is close
to $Q_{{\rm max}}$. First, without specifying a model, we show analytically
that inflation cannot take place in the core of a Q-ball, contrary to the claim
of previous work. Next, for the Affleck-Dine model, we analyze perturbation of
equilibrium solutions with $Q\approx Q_{{\rm max}}$ by numerical analysis of
dynamical field equations. We find that the extremal solution with $Q=Q_{{\rm
max}}$ and unstable solutions around it are "critical solutions", which means
the threshold of black-hole formation.
