  We investigated the time-variability and spectral properties of the eclipsing
X-ray source Circinus Galaxy X-1 (GG X-1), using Chandra, XMM-Newton and ROSAT.
We phase-connected the lightcurves observed over 20 years, and obtained a
best-fitting period $P = (25,970.0 \pm 0.1)$ s $\approx$7.2 hr, and a period
derivative $\dot{P}/P = ( 10.2\pm4.6) \times 10^{-7}$ yr$^{-1}$. The X-ray
lightcurve shows asymmetric eclipses, with sharp ingresses and slow, irregular
egresses. The eclipse profile and duration vary substantially from cycle to
cycle. We show that the X-ray spectra are consistent with a power-law-like
component, absorbed by neutral and ionized Compton-thin material, and by a
Compton-thick, partial-covering medium, responsible for the irregular dips. The
high X-ray/optical flux ratio rules out the possibility that CG X-1 is a
foreground Cataclysmic Variable; in agreement with previous studies, we
conclude that it is the first example of a compact ultraluminous X-ray source
fed by a Wolf-Rayet star or stripped Helium star. Its unocculted luminosity
varies between $\approx$4 $\times 10^{39}$ erg s$^{-1}$ and $\approx$3 $\times
10^{40}$ erg s$^{-1}$. Both the donor star and the super-Eddington compact
object drive powerful outflows: we suggest that the occulting clouds are
produced in the wind-wind collision region and in the bow shock in front of the
compact object. Among the rare sample of Wolf-Rayet X-ray binaries, CG X-1 is
an exceptional target for studies of super-critical accretion and close binary
evolution; it is also a likely progenitor of gravitational wave events.
