  We consider a conformal invariant of braids, the extremal length with totally
real horizontal boundary values $\lambda_{tr}$. The invariant descends to an
invariant of elements of $\mathcal{B}_n\diagup\mathcal{Z}_n$, the braid group
modulo its center. We prove that the number of elements of
$\mathcal{B}_3\diagup\mathcal{Z}_3$ of positive $\lambda_{tr}$ grows
exponentially. The estimate applies to obtain effective finiteness theorems in
the spirit of the geometric Shafarevich conjecture over Riemann surfaces of
second kind. As a corollary we obtain another proof of the exponential growth
of the number of conjugacy classes of $\mathcal{B}_3\diagup\mathcal{Z}_3$ with
positive entropy not exceeding $Y$.
