Your claim that "all solutions from general relativity are geodesically incomplete" is not true. The classical Schwarzschild/Kerr black hole solutions are geodesically incomplete, and along with Minkowski space (which is flat), these are the most well known explicit metrics.
However, many solutions to the (matterless) Einstein equations are geodesically complete. This follows, for example, from the work of Christodoulou-Klainerman http://press.princeton.edu/titles/5159.html, which (very roughly) says that for a sufficiently small appropriate perturbation of $(\mathbb{R}^3,\delta)$, using this as initial data for the Einstein equations yields a Lorentzian Ricci flat manifold which is geodesically complete (and much more: they showed it was asymptotic to Minkowski space in some sense). A nice discussion is given here.
Continuing the theme of physically related Ricci flat metrics, here is an interesting (explicit) Riemannian example:
The Lorentzian Schwarzschild metric (in $4$-dimensions) is incomplete, as you say. However, a strange observation is that formally setting $\tau = it$ (known to the physicists as "Wick rotation") yields a Riemannian manifold which is Ricci flat. The amazing thing is that this metric turns out to be complete, as long as $\tau$ is considered to be periodic with the appropriate period.
This gives a complete Ricci flat metric on $S^2\times\mathbb{R}^2$.
See, for example, section 2 of http://arxiv.org/pdf/hep-th/9112065v1.pdf. Or you can search for "Euclidean black hole" or "Schwarzschild instanton" for more physics literature.