  We derive a simple functional equation with two catalytic variables
characterising the generating function of 3-stack-sortable permutations. Using
this functional equation, we extend the 174-term series to 1000 terms. From
this series, we conjecture that the generating function behaves as $$W(t) \sim
C_0(1-\mu_3 t)^\alpha \cdot \log^\beta(1-\mu_3 t), $$ so that $$[t^n]W(t)=w_n
\sim \frac{c_0\mu_3^n}{ n^{(\alpha+1)}\cdot \log^\lambda{n}} ,$$ where $\mu_3 =
9.69963634535(30),$ $\alpha = 2.0 \pm 0.25.$ If $\alpha = 2$ exactly, then
$\lambda = -\beta+1$, and we estimate $\beta \approx -3.$ If $\alpha$ is not an
integer, then $\lambda=-\beta$, but we cannot give a useful estimate of
$\beta$. The growth constant estimate (just) contradicts a conjecture of the
first author that $$9.702 < \mu_3 \le 9.704.$$ We also prove a new rigorous
lower bound of $\mu_3\geq 9.4854$, allowing us to disprove a conjecture of
B\'ona.
  We then further extend the series using differential-approximants to obtain
approximate coefficients $O(t^{2000}),$ expected to be accurate to $20$
significant digits, and use the approximate coefficients to provide additional
evidence supporting the results obtained from the exact coefficients.
