  We introduce the concept of {\em maximal lineability cardinal number},
$\mL(M)$, of a subset $M$ of a topological vector space and study its relation
to the cardinal numbers known as: additivity $A(M)$, homogeneous lineability
$\HL(M)$, and lineability $\LL(M)$ of $M$. In particular, we will describe, in
terms of $\LL$, the lineability and spaceability of the families of the
following Darboux-like functions on $\real^n$, $n\ge 1$: extendable, Jones, and
almost continuous functions.
