Answer
$18278$
Work Step by Step
$n$ objects in $r$ spots where each spot can contain any of the objects is $n^r$.
Thus the number of $1$ letter long company names: $26^1=26$
Thus the number of $2$ letter long company names: $26^2=676$
Thus the number of $3$ letter long company names: $26^3=17576$
Thus in total: $26+676+17576=18278$
