Finite-field many-body perturbation theory and coupled cluster calculations are reported for the static second dipole hyperpolarizability gamma(alpha beta gamma delta) of trans-butadiene. A very large basis set of [9s6p4d1f/6s3p1d] size (336 contracted Gaussian-type functions) should lead to self-consistent field (SCF) values of near-Hartree-Fock quality. We report gamma(xxxx) = 6.19, gamma(xxxz) = -0.44, gamma(xxyy) = 3.42, gamma(zzxx) = 2.07, gamma(xyyz) = -0.50, gamma(xzzz) = 1.73, gamma(yyyy) = 14.72, gamma(yyzz) = 8.46, gamma(zzzz) = 24.10 and <(gamma)over bar> = 14.58 for 10(-3) x gamma(alpha beta gamma delta)/e(4)a(0)(4)E(h)(-3) at the experimental geometry (molecule on the xz plane with z as the main axis). <(gamma)over bar> = (14.6+/-0.4) x 10(3) e(4)a(0)(4)E(h)(-3) should be a very reliable estimate of the Hartree-Fock limit of the mean hyperpolarizability. Keeping all other molecular geometry parameters constant, we find that near the Hartree-Fock limit the mean hyperpolarizability varies with the C=C bond length as 10(-3) x <(gamma)over bar>(R-C=C)/e(4)a(0)(4)E(h)(-3) = 14.93+31.78 Delta R+30.88 Delta R-2-2.96 Delta R-3 and with the C-C bond length as 10(-3) x <(gamma)over bar>(RC-C)/e(4)a(0)(4)E(h)(-3) = 14.93-7.20 Delta R+3.04 Delta R-2, where Delta R/a(0) is the displacement from the respective experimental value. The dependence of the components of gamma(alpha beta gamma delta) on the molecular geometry parameters is not uniform. Electron correlation corrections have been calculated at various molecular geometries at the coupled-cluster single, double and perturbatively linked triple excitations level of theory for all independent components of gamma(alpha beta gamma delta). In absolute terms, electron correlation affects strongly the gamma(zzzz), less strongly the gamma(xxxx), and even less strongly the out-of-plane component gamma(yyyy). The present analysis suggests a conservative estimate of (3.0+/-0.6) x 10(3) e(4)a(0)(4)E(h)(-3) for the electron correlation correction to <(gamma)over bar> at the experimental molecular geometry. Most of this value is appropriate to gamma(zzzz). A static limit of <(gamma)over bar> = (17.6+/-1.0) x 10(3) e(4)a(0)(4)E(h)(-3) is advanced (neglecting vibrational averaging). Even if a crude theoretical estimate of the dispersion of <(gamma)over bar> at 1064 nm is added to this value, the result sets up an unambiguous claim to accord with the experimental value of (20.18+/-0.11) x 10(3) e(4)a(0)(4)E(h)(-3) [D. P. Shelton, Phys. Rev. A 42, 2578 (1990)].