  The large-matrix limit laws of the rescaled largest eigenvalue of the
orthogonal, unitary and symplectic $n$-dimensional Gaussian ensembles -- and of
the corresponding Laguerre ensembles (Wishart distributions) for various
regimes of the parameter $\alpha$ (degrees of freedom $p$) -- are known to be
the Tracy-Widom distributions $F_\beta$ ($\beta=1,2,4$). We will establish
(paying particular attention to large, or small, ratios $p/n$) that, with
careful choices of the rescaling constants and the expansion parameter $h$, the
limit laws embed into asymptotic expansions in powers of $h$, where $h \asymp
n^{-2/3}$ resp. $h \asymp (n\,\wedge\,p)^{-2/3}$. We find explicit analytic
expressions of the first few expansions terms as linear combinations, with
rational polynomial coefficients, of higher order derivatives of the limit law
$F_\beta$. With a proper parametrization, the expansions in the Gaussian cases
can be understood, for given $n$, as the limit $p\to\infty$ of the Laguerre
cases. Whereas the results for $\beta=2$ are presented with proof, the
discussion of the cases $\beta=1,4$ is based on some hypotheses, focussing on
the algebraic aspects of actually computing the polynomial coefficients. For
the purposes of illustration and validation, the various results are checked
against simulation data with large sample sizes.
