The configuration interaction (CI) method using a large Laguerre basis restricted to ℓ = 0 orbitals is applied to the calculation of the He ground state. The maximum number of orbitals included was 60. The numerical evidence suggests that the energy converges as ΔEN ≈ A/N7/2 + B/N8/2 + …, where N is the number of Laguerre basis functions. The electron–electron δ‐function expectation converges as ΔδN ≈ A/N5/2 + B/N6/2 + …, and the variational limit for the ℓ = 0 basis is estimated as 0.1557637174(2) a(0)(3). It was seen that extrapolation of the energy to the variational limit is dependent on the basis dimension at which the exponent in the Laguerre basis was optimized. In effect, it may be best to choose a nonoptimal exponent if one wishes to extrapolate to the variational limit. An investigation of the natural orbital asymptotics revealed the energy converged as ΔEN ≈ A/N6 + B/N7 + …, while the electron–electron δ‐function expectation converged as ΔδN ≈ A/N4 + B/N5 + …. The asymptotics of expectation values other than the energy showed fluctuations that depended on whether N was even or odd.