  A function which is transcendental and meromorphic in the plane has at least
two singular values. On one hand, if a meromorphic function has exactly two
singular values, it is known that the Hausdorff dimension of the escaping set
can only be either $2$ or $1/2$. On the other hand, the Hausdorff dimension of
escaping sets of Speiser functions can attain every number in $[0,2]$ (cf.
\cite{ac1}). In this paper, we show that number of singular values which is
needed to attain every Hausdorff dimension of escaping sets is not more than
$4$.
