—   genus ($g\geq1$)

For the rows held here, $\delta_g^+$ is the largest real root of $P_g(x)$. The polynomials are $P_1=x^2-3x+1$, $P_2=x^4-x^3-x^2-x+1$, $P_3=x^6-x^4-x^3-x^2+1$, $P_4=x^8-x^5-x^4-x^3+1$, $P_5=x^{10}+x^9-x^7-x^6-x^5-x^4-x^3+x+1$, $P_7=x^{14}+x^{13}-x^9-x^8-x^7-x^6-x^5+x+1$, and $P_8=x^{16}-x^9-x^8-x^7+1$ [1].

The topological entropy of a pseudo-Anosov homeomorphism with dilatation $\lambda$ is $\log\lambda$ [1].

$\delta_g^+$ is taken over a restricted class, so it is at least the unrestricted minimum $\delta_g$ over all pseudo-Anosov homeomorphisms of $\Sigma_g$. The two are equal for $g=1$ and $g=2$ [1], so those rows are also $\delta_1$ and $\delta_2$.

For $2\leq g\leq5$, the values are Salem numbers and are largest roots of the degree $2g$ polynomials listed in (1). The rows $g=4$ and $g=5$ are less than $1.3$ and also occur in the table of Salem numbers less than $1.3$. The values for $g=7,8$ are reciprocal Perron numbers but not Salem numbers: for each such $g$, $P_g$ has exactly three roots outside the unit circle, including $\delta_g^+$ itself.

Lanneau and Thiffeault prove $\delta_6^+\geq\delta_5^+$, so the orientable minimum dilatations are not strictly decreasing with genus [1]. There is no $g=6$ row because $\delta_6^+$ is not known; Lanneau and Thiffeault's inequality is only a lower bound [5]. The known values also have $\delta_7^+<\delta_8^+$.

Sage

from sage.all import PolynomialRing, QQ, RealField

RR = RealField(400)
R = PolynomialRing(QQ, 'x')
x = R.gen()
polynomials = {
    1: x**2 - 3*x + 1,
    2: x**4 - x**3 - x**2 - x + 1,
    3: x**6 - x**4 - x**3 - x**2 + 1,
    4: x**8 - x**5 - x**4 - x**3 + 1,
    5: x**10 + x**9 - x**7 - x**6 - x**5 - x**4 - x**3 + x + 1,
    7: x**14 + x**13 - x**9 - x**8 - x**7 - x**6 - x**5 + x + 1,
    8: x**16 - x**9 - x**8 - x**7 + 1,
}
print({g: max(p.roots(RR, multiplicities=False))
       for g, p in polynomials.items()})

Erwan Lanneau and Jean-Luc Thiffeault, On the minimum dilatation of pseudo-Anosov homeomorphisms on surfaces of small genus, Annales de l'Institut Fourier 61 (2011), 105-144. (arXiv) (doi)

John William Aaber and Nathan M. Dunfield, Closed surface bundles of least volume, Algebraic & Geometric Topology 10 (2010), 2315-2342. (arXiv) (doi)

Eiko Kin and Mitsuhiko Takasawa, Pseudo-Anosovs on closed surfaces having small entropy and the Whitehead sister link exterior, Journal of the Mathematical Society of Japan 65 (2013), no. 2, 411-446. (arXiv) (doi)

Eriko Hironaka, Small dilatation pseudo-Anosov mapping classes coming from the simplest hyperbolic braid, Algebraic & Geometric Topology 10 (2010), 2041-2060. (arXiv) (doi)

Eriko Hironaka, Small dilatation pseudo-Anosov mapping classes and short circuits on train track automata, Institut Mittag-Leffler preprint series, 2014. (arXiv)

Wikipedia: Pseudo-Anosov map

Growth rates of hyperbolic Coxeter triangle groups —   its growth rates include every value here from $g=2$ to $g=5$, indexed by Coxeter diagram rather than by surface genus

Salem numbers less than $1.3$ —   holds the same Salem numbers as the rows $g=4$ and $g=5$, indexed by their minimal polynomials rather than by genus

Lyapunov exponents of classical chaotic systems —   holds the Lyapunov exponent of Arnold's cat map, which is $\log\delta_1^+$

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it holds the exact values established in the cited small-genus results: $1\leq g\leq5$ and $g=7,8$; genus $6$ is open: Lanneau and Thiffeault's lower bound for genus $6$ is $\delta_5^+$ itself [1], and Hironaka's survey states that the exact value is not known [5])

How they were obtained:

Genus $1$ is the torus case recalled by Lanneau and Thiffeault, $\delta_2^+$ is Theorem 1.1, and $\delta_3^+,\ldots,\delta_5^+$ are Theorem 1.2 of [1].

more

The genus $7$ and $8$ lower bounds are Theorem 1.3 of [1]; the genus $7$ bound is realized by Aaber and Dunfield [2] and independently by Kin and Takasawa [3], and the genus $8$ bound is realized by Hironaka [4]. The generator isolates the real root greater than $1$ of each exact polynomial in interval arithmetic and verifies that the polynomial changes sign across the isolating interval before writing $100$ digits.