  A new C*-enlargement of a C*-algebra $A$ nested between the local multiplier
algebra $M_{\text{loc}}(A)$ of $A$ and its injective envelope $I(A)$ is
introduced. Various aspects of this maximal C*-algebra of quotients,
$Q_{\text{max}}(A)$, are studied, notably in the setting of AW*-algebras. As a
by-product we obtain a new example of a type I C*-algebra $A$ such that
$M_{\text{loc}}(M_{\text{loc}}(A))\ne M_{\text{loc}}(A)$.
