Complete edit, after talking to a colleague -
Suppose that $G$ is the mapping class group of a surface $S$. Then you are asking:
Is there a number $K$, depending only on $S$, with the following property? For every $\sigma \in G$ there are torsion elements $\tau_i \in G$ so that $\sigma = \Pi_{i = 1}^K \tau_i$.
As Henry Wilton puts it - you are asking if the mapping class group is boundedly generated by torsion. The answer to this question is "No". This follows from a paper of Bestvina and Fujiwara "Bounded cohomology of subgroups of mapping class groups". Basically, our They show that the group $G$ admits unbounded quasi-homomorphisms. I can provide further details, but See the first five pages of their paperreally does .
Edit - to give you what you wanta few details. A $D$-quasi-homomorphism is a map $\phi$ from $G$ to the reals so that for all $g,f \in G$ we have $|\phi(gf) - \phi(g) - \phi(f)| < D$. It is an exercise to show that if $g$ is torsion then $|\phi(g)| < D$. Thus, if $G$ was boundedly generated, say with constant $K$, then we would have, for all $g \in G$, that $|\phi(g)| < 2KD$. This is a contradiction.
