The static and frequency-dependent hyperpolarizability of Ne is calculated using the coupled cluster approximate triples model CC3 and a large correlation-consistent basis set. Relativisitic contributions are taken into account through first-order. Analyzing in details the one- and N-electron errors, the uncertainty of the results is estimated to be about one atomic unit, an accuracy which for the first time is obtained for a many electron system in ab initio calculations of frequency-dependent second hyperpolarizabilities. For the static hyperpolarizability of Ne we obtain a best theoretical estimate of 107.6 +/- 1.0 a.u. The frequency-dependent hyperpolarizability of Ne is also calculated at the lower levels of the coupled cluster hierarchy -CCS, CC2, and CCSD- and compared to M to illustrate the importance of the inclusion of the triple excitations in coupled cluster wavefunctions for an accurate determination of hyperpolarizabilities. The results at the CC3 level agree well with the latest experimental results. (C) 2004 Elsevier B.V. All rights reserved.