Abstract
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 $\beta \approx -2,$ but with a wide uncertainty of $\pm 1.$ 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 Bóna. 
 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.
Highlights
A permutation is said to be stack-sortable if, when it is passed through a stack, the result is the increasing permutation
The exponent estimates are not stable and are decreasing with the number of terms. This is characteristic of a generating function with a confluent logarithmic term, and we suggest that these exponent the electronic journal of combinatorics 28(2) (2021), #P2.49
We suggest that the same phenomenon is happening with the generating function for 3-stack-sortable permutations and that the correct exponent value is more likely to be the electronic journal of combinatorics 28(2) (2021), #P2.49 around 2.0, with a confluent logarithmic term
Summary
A permutation is said to be stack-sortable if, when it is passed through a stack, the result is the increasing permutation. A permutation is called k-stack-sortable if iteratively applying the stack-sorting map to it k times results in the increasing permutation. We start by reinterpreting these arguments to derive a functional equation that characterises this generating function Using this functional equation, we write an efficient algorithm with which we compute the numbers wn for n 1000. We use the differential-approximants to (approximately) extend the series from 1000 terms to 2000 terms, using a method of series extension developed by the third author [18], where the newly-obtained coefficients are expected to be accurate to at least 20 significant digits This is more than sufficient for ratio-based analysis methods, so we re-analyse the extended series
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