  Given a family $F$ of pairwise almost disjoint sets on a countable set $S$,
we study maximal almost disjoint (mad) families $F^+$ extending $F$.
  We define $a^+(F)$ to be the minimal possible cardinality of $F^+\setminus F$
for such $F^+$, and $a^+(\kappa)=\sup\{a^+(F): |F| \leq \kappa \}$. We show
that all infinite cardinal less than or equal to the continuum continuum can be
represented as $a^+(F)$ for some almost disjoint $F$ and that the inequalities
$\aleph_1=a<a^+(\aleph_1)=c$ and $a=a^+(\aleph_1)<c$ are both consistent.
  We also give a several constructions of mad families with some additional
properties.

Logic